Ka-Kyung Kim
2015년 4월 29일
- 베이지안 통계 R
- Doing Bayesian Data Analysis, Second Edition: A Tutorial with R, JAGS, and Stan
- Source Code
- Exercises and solutions
- Latex
- 9.1. A Single Coin from a Single Mint
- 9.2. Multiple Coins from a Single Mint
- 9.3. Shrinkage in Hierarchical Models
- 9.4. Speeding up JAGS
- 9.5. Extending the Hierarchy: Subjects Within Categories
Hierarchical model?
Mathematical descriptions involving multiple parameters such that the credible values of some parameters meaningfully depend on the values of other parameters.
example) coins minted from the same factory.
Credible values of
examples) batting ability ofbaseball players who have different primary fielding positions, probability that a child bought lunch from the school cafeteria, in different schools in different districts, probability that patients recover from heart surgery performed by different surgical teams within different hospitals in different cities, etc.
The parameters at different levels in a hierarchical model coexist in a joint parameter space.
The data depend only on the value of
The dependencies among parameters become useful.
- For the given application.
- All the data can jointly inform all the parameter estimates.
- The dependencies can facilitate efficient Monte Carlo sampling from the posterior distribution, because clever algorithms can take advantage of conditional distributions (ex: Gibbs sampling uses conditional distributions).
Real-world data involving dichotomous outcomes
-
Coin flip
-
Recovery or nonrecovery from a disease after treatment
-
Recalling or forgetting a studied item
-
Choosing candidate A or candidate B in an election
-
Therapeutic touch
-
Extrasensory perception
-
Baseball batting ability
The likelihood function is the Bernoulli distribution.
$$ \gamma_{i} \sim dbern(\theta)\quad (9.2) $$
The prior distribution is a beta density.
$$ \theta\sim dbeta(a, b) \quad (9.3) $$
Reference in Chapter 6 (p129)
Our goal is to convert a prior belief expressed in terms of central tendency and sample size into equivalent values of a and b in the beta distribution.
Toward this goal, it is useful to know the central tendency and spread of the beta distribution expressed in terms of a and b.
It turns out that the mean of the beta(
Thus, when a = b, the mean and mode are 0.5. When a > b, the mean and mode are greater than 0.5, and when a < b, the mean and mode are less than 0.5.
The spread ofthe beta distribution is related to the "concentration" K = a+ b. You can see from Figure 6.1 that as K = a + b gets larger, the beta distribution gets narrower or more concentrated.
The shape parameters of the beta density, a and b, can be re-expressed in terms of the mode
$$ \theta \sim dbeta (\omega(K-2)+1, (1-\omega)(K-2)+1) \quad (9.4) $$
The larger K is, the more consistently the mint makes coins with
The three terms of the numerator in second line
- The likelihood function is a Bernoulli distribution in Equation 9.2
- The dependence of
$\theta$ on$\omega$ is specified to be a beta density in Equation 9.4. - The prior distribution on
$\omega$ is a beta density in Equation 9.5.
The likelihood and prior can be re-expressed as a hierarchical chain of dependencies among parameters
- Important in
- Human interpretation
- Hierarchical models can be parsed for MCMC samplers very effectively
- Applications
- Meaningful chain of dependencies is primary in the conceptual genesis of the model
- Thinking of the model as a joint distribution on a joint parameter space
When the parameters extend over a finite domain, and there are not too many of them, then we can approximate the posterior via grid approximation.
The prior distribution has
As this is a grid approximation,
- The joint prior p(
$\theta$ ,$\omega$ ) = p($\theta$ ,$\omega$ ) * p($\omega$ ) at every grid point - Dividing by their sum across the entire grid.
- The normalized probability masses -> (converted to) -> estimates of probability density at each grid point
by dividing each probability mass by the area of a grid cell.
prior p(
Impact on beliefs about how
High prior certainty regarding
Figure 9.3 illustrates such a case, where
The likelihood graphs look the same in the two figures.
The contour lines are parallel to the
Comparison
In Figure 9.2
- Low prior certainty on
$\omega$ (small$A\omega$ ,$B\omega$) - High prior certainty about the dependence of
$\theta$ on$\omega$ (large$K$ ). - The posterior distribution showed a big change in beliefs about
$\omega$ , but not much change in beliefs about the dependence of$\theta$ on$\omega$ .
In Figure 9.3
- Complementary situation
- High prior certainty on
$\omega$ and low prior certainty about the dependence of$\theta$ on$\omega$ .
The grid approximation displayed in Figures 9.5 and 9.6 used combs of only 50 points on each parameter (
In Figure 9.6, the prior on
but the prior dependency of
This shift away from the data proportion of subject 2 is caused by the fact that the other coin had a larger sample size
=>
The values of the prior and likelihood at a large number of grid points and sums over them to determine the denominator of Bayes' rule.
Grid approximation is not a viable approach to even modestly large problems.
When K of the concentration or consistency parameter was fixed at a large value, the individual
When K was fixed at a small value,
=> In real situations, we do not know the value of K in advance!
The gamma(K|s,r) distribution is a probability density for k ≥ 0, with two parameters that determine its exact form, called its shape s and rate r parameters.
Prior beliefs are most intuitively expressed in terms of the central tendency and a width of the distribution.
Mean of the gamma distribution:
Mode of
Standard deviation
#install.packages("knitr")
library(knitr)## Warning: package 'knitr' was built under R version 3.1.3
#install.packages('rjags')
require(rjags)## Loading required package: rjags
## Warning: package 'rjags' was built under R version 3.1.3
## Loading required package: coda
## Warning: package 'coda' was built under R version 3.1.3
## Linked to JAGS 3.4.0
## Loaded modules: basemod,bugs
#setwd("D:/Study/Bayesian/bayesianR/ch09")
source("DBDA2E-utilities.R")##
## *********************************************************************
## Kruschke, J. K. (2015). Doing Bayesian Data Analysis, Second Edition:
## A Tutorial with R, JAGS, and Stan. Academic Press / Elsevier.
## *********************************************************************
## Warning: package 'runjags' was built under R version 3.1.3
gammaShRaFromMeanSD(mean=10, sd=100)## $shape
## [1] 0.01
##
## $rate
## [1] 0.001
gammaShRaFromModeSD(mode=10, sd=100)## $shape
## [1] 1.105125
##
## $rate
## [1] 0.01051249
gammaParam = gammaShRaFromModeSD(mode=10, sd=100)
gammaParam$shape## [1] 1.105125
gammaParam$rate## [1] 0.01051249
The arrows in the hierarchical diagram indicate the dependencies between the variables. Every arrow in the diagram has a corresponding expression in the JAGS model specification.
# Define the model:
#modelString = "
#model {
# for ( i in 1:Ntotal ) {
# y[i] ~ dbern( theta )
# }
# theta ~ dbeta( 1 , 1 )
#}
#model {
# for ( i 1n l:Ntotal ) {
# y[i] ~ dbern( theta[s[i]])
# }
# for ( s in 1:Nsubj ) {
# theta[s] - dbeta( omega*(kappa-2)+1, (1-omega)*(kappa-2)+1 )
# }
# omega ~ dbeta( l , l )
# kappa <- kappaMinusTwo + 2
# kappaMinusTwo - dgamma( 0.01 , 0.01 ) # mean=1 , sd=10 (generic vague)
#}Therapeutic touch is a nursing technique in which the practitioner manually manipulates the "energy field" of a patient who is suffering from a disease. (Rosa, Rosa, Sarner, and Barrett (1998)) Each trial by flip of the coin -> The practitioner guess -> Each trial was scored as correct or wrong.
- Each 21 practitioner for 10 trials
- 7 practitioner tested twice approximately a year apart
- 30 paramters
=> How much the group as a whole or any individuals differed from chance performance(=0.50)?
Contents of 'TherapeuticTouchData.csv'
y: outcome of the trials (0 or 1)
s: identifier of the subject who generated each outcome
s Name, y Name in the functions genMCMC and plotMCMC
#Read the data file:
myData = read.csv("TherapeuticTouchData.csv")
# Load the relevant model functions into R's working memory:
source("Jags-Ydich-XnomSsubj-MbernBetaOmegaKappa.R")##
## *********************************************************************
## Kruschke, J. K. (2015). Doing Bayesian Data Analysis, Second Edition:
## A Tutorial with R, JAGS, and Stan. Academic Press / Elsevier.
## *********************************************************************
# Generate the MCMC chain:
mcmcCoda = genMCMC(data=myData, sName="s", yName="y",
numSavedSteps=20000 , thinSteps=10)## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph Size: 602
##
## Initializing model
##
## Burning in the MCMC chain...
## Sampling final MCMC chain...
#Display diagnostics of chain, for specified parameters:
diagMCMC (codaObject=mcmcCoda, parName="omega")
Output 9.1
diagMCMC (codaObject=mcmcCoda, parName="kappa")diagMCMC (codaObject=mcmcCoda, parName="theta[1]")# Get summary statistics of chain:
# diffIdVec: vector of subject indices that should have their posterior differences summarized.
smryMCMC (mcmcCoda, compVal=0.5, diffIdVec=c(1,14,28), compValDiff=0.0)## Mean Median Mode ESS HDImass
## omega 0.43556782 0.43595107 0.43863404 12485.1 0.95
## kappa 54.88247592 36.71597983 19.38930965 3868.5 0.95
## theta[1] 0.35947618 0.36775328 0.39106132 12252.8 0.95
## theta[2] 0.38290978 0.38819523 0.40824207 13782.1 0.95
## theta[3] 0.40808607 0.41142332 0.42087586 16618.8 0.95
## theta[4] 0.40774125 0.41085717 0.41902959 16480.7 0.95
## theta[5] 0.40670969 0.40986798 0.41229704 15256.3 0.95
## theta[6] 0.40741710 0.41077965 0.42691357 15724.1 0.95
## theta[7] 0.40688779 0.40954804 0.41525995 14850.7 0.95
## theta[8] 0.40661922 0.40981455 0.41493553 16329.5 0.95
## theta[9] 0.40667966 0.40945389 0.41238421 16775.5 0.95
## theta[10] 0.40755803 0.41099475 0.42203032 16917.2 0.95
## theta[11] 0.43075949 0.43110783 0.44034987 18747.5 0.95
## theta[12] 0.43120807 0.43061108 0.42368678 18512.4 0.95
## theta[13] 0.43037570 0.42997653 0.42718119 19091.3 0.95
## theta[14] 0.43090708 0.43054336 0.42104599 19081.7 0.95
## theta[15] 0.43026789 0.43057605 0.43433190 17819.6 0.95
## theta[16] 0.45485626 0.45209924 0.44310134 17927.9 0.95
## theta[17] 0.45472889 0.45209440 0.45222730 18562.5 0.95
## theta[18] 0.45427452 0.45114123 0.44118825 19350.0 0.95
## theta[19] 0.45423319 0.45164476 0.44096249 18160.8 0.95
## theta[20] 0.45436521 0.45191410 0.44888229 18472.4 0.95
## theta[21] 0.45311467 0.45081189 0.44967627 19010.9 0.95
## theta[22] 0.45373573 0.45097217 0.44760526 18534.4 0.95
## theta[23] 0.47787249 0.47144717 0.45940378 16984.5 0.95
## theta[24] 0.47769739 0.47211016 0.46420290 16168.7 0.95
## theta[25] 0.50144704 0.49274640 0.47937726 14133.5 0.95
## theta[26] 0.50194704 0.49364271 0.48792257 14131.7 0.95
## theta[27] 0.50167759 0.49274863 0.47907924 14922.5 0.95
## theta[28] 0.52522486 0.51437938 0.48887439 12408.0 0.95
## theta[1]-theta[14] -0.07143090 -0.06095796 -0.04593949 16398.4 0.95
## theta[1]-theta[28] -0.16574868 -0.14454902 -0.09532450 10409.5 0.95
## theta[14]-theta[28] -0.09431777 -0.07938548 -0.05196311 14225.1 0.95
## HDIlow HDIhigh CompVal PcntGtCompVal ROPElow
## omega 0.3634888 0.50896450 0.5 3.855 NA
## kappa 4.2500889 162.71733768 NA NA NA
## theta[1] 0.1758237 0.51950356 0.5 3.770 NA
## theta[2] 0.2128656 0.53807904 0.5 6.560 NA
## theta[3] 0.2391448 0.56240266 0.5 11.510 NA
## theta[4] 0.2480878 0.56450251 0.5 11.345 NA
## theta[5] 0.2430916 0.56377919 0.5 11.400 NA
## theta[6] 0.2424339 0.56265368 0.5 11.470 NA
## theta[7] 0.2438985 0.56587793 0.5 11.325 NA
## theta[8] 0.2423211 0.56076931 0.5 11.130 NA
## theta[9] 0.2410394 0.56178020 0.5 11.275 NA
## theta[10] 0.2418789 0.56123887 0.5 11.505 NA
## theta[11] 0.2666391 0.58226506 0.5 17.975 NA
## theta[12] 0.2706876 0.59073060 0.5 18.270 NA
## theta[13] 0.2715880 0.59036785 0.5 18.115 NA
## theta[14] 0.2748292 0.59448117 0.5 17.880 NA
## theta[15] 0.2639835 0.58132072 0.5 17.475 NA
## theta[16] 0.2931170 0.60937511 0.5 26.465 NA
## theta[17] 0.3011809 0.62293185 0.5 26.770 NA
## theta[18] 0.2986064 0.61824352 0.5 26.450 NA
## theta[19] 0.2962029 0.61351935 0.5 26.285 NA
## theta[20] 0.2955589 0.61288103 0.5 26.475 NA
## theta[21] 0.2957731 0.61433654 0.5 26.010 NA
## theta[22] 0.2991205 0.61567599 0.5 26.360 NA
## theta[23] 0.3237553 0.65502282 0.5 36.440 NA
## theta[24] 0.3191052 0.64899318 0.5 36.140 NA
## theta[25] 0.3428332 0.68543635 0.5 46.515 NA
## theta[26] 0.3464274 0.68474784 0.5 46.715 NA
## theta[27] 0.3444499 0.68642292 0.5 46.510 NA
## theta[28] 0.3546349 0.71122678 0.5 56.580 NA
## theta[1]-theta[14] -0.2986243 0.13362958 0.0 25.970 NA
## theta[1]-theta[28] -0.4564162 0.06170424 0.0 8.410 NA
## theta[14]-theta[28] -0.3569836 0.11044988 0.0 20.420 NA
## ROPEhigh PcntLtROPE PcntInROPE PcntGtROPE
## omega NA NA NA NA
## kappa NA NA NA NA
## theta[1] NA NA NA NA
## theta[2] NA NA NA NA
## theta[3] NA NA NA NA
## theta[4] NA NA NA NA
## theta[5] NA NA NA NA
## theta[6] NA NA NA NA
## theta[7] NA NA NA NA
## theta[8] NA NA NA NA
## theta[9] NA NA NA NA
## theta[10] NA NA NA NA
## theta[11] NA NA NA NA
## theta[12] NA NA NA NA
## theta[13] NA NA NA NA
## theta[14] NA NA NA NA
## theta[15] NA NA NA NA
## theta[16] NA NA NA NA
## theta[17] NA NA NA NA
## theta[18] NA NA NA NA
## theta[19] NA NA NA NA
## theta[20] NA NA NA NA
## theta[21] NA NA NA NA
## theta[22] NA NA NA NA
## theta[23] NA NA NA NA
## theta[24] NA NA NA NA
## theta[25] NA NA NA NA
## theta[26] NA NA NA NA
## theta[27] NA NA NA NA
## theta[28] NA NA NA NA
## theta[1]-theta[14] NA NA NA NA
## theta[1]-theta[28] NA NA NA NA
## theta[14]-theta[28] NA NA NA NA
## Mean Median Mode ESS HDImass
## omega 0.43556782 0.43595107 0.43863404 12485.1 0.95
## kappa 54.88247592 36.71597983 19.38930965 3868.5 0.95
## theta[1] 0.35947618 0.36775328 0.39106132 12252.8 0.95
## theta[2] 0.38290978 0.38819523 0.40824207 13782.1 0.95
## theta[3] 0.40808607 0.41142332 0.42087586 16618.8 0.95
## theta[4] 0.40774125 0.41085717 0.41902959 16480.7 0.95
## theta[5] 0.40670969 0.40986798 0.41229704 15256.3 0.95
## theta[6] 0.40741710 0.41077965 0.42691357 15724.1 0.95
## theta[7] 0.40688779 0.40954804 0.41525995 14850.7 0.95
## theta[8] 0.40661922 0.40981455 0.41493553 16329.5 0.95
## theta[9] 0.40667966 0.40945389 0.41238421 16775.5 0.95
## theta[10] 0.40755803 0.41099475 0.42203032 16917.2 0.95
## theta[11] 0.43075949 0.43110783 0.44034987 18747.5 0.95
## theta[12] 0.43120807 0.43061108 0.42368678 18512.4 0.95
## theta[13] 0.43037570 0.42997653 0.42718119 19091.3 0.95
## theta[14] 0.43090708 0.43054336 0.42104599 19081.7 0.95
## theta[15] 0.43026789 0.43057605 0.43433190 17819.6 0.95
## theta[16] 0.45485626 0.45209924 0.44310134 17927.9 0.95
## theta[17] 0.45472889 0.45209440 0.45222730 18562.5 0.95
## theta[18] 0.45427452 0.45114123 0.44118825 19350.0 0.95
## theta[19] 0.45423319 0.45164476 0.44096249 18160.8 0.95
## theta[20] 0.45436521 0.45191410 0.44888229 18472.4 0.95
## theta[21] 0.45311467 0.45081189 0.44967627 19010.9 0.95
## theta[22] 0.45373573 0.45097217 0.44760526 18534.4 0.95
## theta[23] 0.47787249 0.47144717 0.45940378 16984.5 0.95
## theta[24] 0.47769739 0.47211016 0.46420290 16168.7 0.95
## theta[25] 0.50144704 0.49274640 0.47937726 14133.5 0.95
## theta[26] 0.50194704 0.49364271 0.48792257 14131.7 0.95
## theta[27] 0.50167759 0.49274863 0.47907924 14922.5 0.95
## theta[28] 0.52522486 0.51437938 0.48887439 12408.0 0.95
## theta[1]-theta[14] -0.07143090 -0.06095796 -0.04593949 16398.4 0.95
## theta[1]-theta[28] -0.16574868 -0.14454902 -0.09532450 10409.5 0.95
## theta[14]-theta[28] -0.09431777 -0.07938548 -0.05196311 14225.1 0.95
## HDIlow HDIhigh CompVal PcntGtCompVal ROPElow
## omega 0.3634888 0.50896450 0.5 3.855 NA
## kappa 4.2500889 162.71733768 NA NA NA
## theta[1] 0.1758237 0.51950356 0.5 3.770 NA
## theta[2] 0.2128656 0.53807904 0.5 6.560 NA
## theta[3] 0.2391448 0.56240266 0.5 11.510 NA
## theta[4] 0.2480878 0.56450251 0.5 11.345 NA
## theta[5] 0.2430916 0.56377919 0.5 11.400 NA
## theta[6] 0.2424339 0.56265368 0.5 11.470 NA
## theta[7] 0.2438985 0.56587793 0.5 11.325 NA
## theta[8] 0.2423211 0.56076931 0.5 11.130 NA
## theta[9] 0.2410394 0.56178020 0.5 11.275 NA
## theta[10] 0.2418789 0.56123887 0.5 11.505 NA
## theta[11] 0.2666391 0.58226506 0.5 17.975 NA
## theta[12] 0.2706876 0.59073060 0.5 18.270 NA
## theta[13] 0.2715880 0.59036785 0.5 18.115 NA
## theta[14] 0.2748292 0.59448117 0.5 17.880 NA
## theta[15] 0.2639835 0.58132072 0.5 17.475 NA
## theta[16] 0.2931170 0.60937511 0.5 26.465 NA
## theta[17] 0.3011809 0.62293185 0.5 26.770 NA
## theta[18] 0.2986064 0.61824352 0.5 26.450 NA
## theta[19] 0.2962029 0.61351935 0.5 26.285 NA
## theta[20] 0.2955589 0.61288103 0.5 26.475 NA
## theta[21] 0.2957731 0.61433654 0.5 26.010 NA
## theta[22] 0.2991205 0.61567599 0.5 26.360 NA
## theta[23] 0.3237553 0.65502282 0.5 36.440 NA
## theta[24] 0.3191052 0.64899318 0.5 36.140 NA
## theta[25] 0.3428332 0.68543635 0.5 46.515 NA
## theta[26] 0.3464274 0.68474784 0.5 46.715 NA
## theta[27] 0.3444499 0.68642292 0.5 46.510 NA
## theta[28] 0.3546349 0.71122678 0.5 56.580 NA
## theta[1]-theta[14] -0.2986243 0.13362958 0.0 25.970 NA
## theta[1]-theta[28] -0.4564162 0.06170424 0.0 8.410 NA
## theta[14]-theta[28] -0.3569836 0.11044988 0.0 20.420 NA
## ROPEhigh PcntLtROPE PcntInROPE PcntGtROPE
## omega NA NA NA NA
## kappa NA NA NA NA
## theta[1] NA NA NA NA
## theta[2] NA NA NA NA
## theta[3] NA NA NA NA
## theta[4] NA NA NA NA
## theta[5] NA NA NA NA
## theta[6] NA NA NA NA
## theta[7] NA NA NA NA
## theta[8] NA NA NA NA
## theta[9] NA NA NA NA
## theta[10] NA NA NA NA
## theta[11] NA NA NA NA
## theta[12] NA NA NA NA
## theta[13] NA NA NA NA
## theta[14] NA NA NA NA
## theta[15] NA NA NA NA
## theta[16] NA NA NA NA
## theta[17] NA NA NA NA
## theta[18] NA NA NA NA
## theta[19] NA NA NA NA
## theta[20] NA NA NA NA
## theta[21] NA NA NA NA
## theta[22] NA NA NA NA
## theta[23] NA NA NA NA
## theta[24] NA NA NA NA
## theta[25] NA NA NA NA
## theta[26] NA NA NA NA
## theta[27] NA NA NA NA
## theta[28] NA NA NA NA
## theta[1]-theta[14] NA NA NA NA
## theta[1]-theta[28] NA NA NA NA
## theta[14]-theta[28] NA NA NA NA
# Display posterior information:
plotMCMC (mcmcCoda, data=myData, sName="s", yName="y",
compVal=0.5, diffIdVec=c(1,14,28), compValDiff=0.0)
Output 9.2: same as Figure 9.10 upper pannel
Output 9.3: same as Figure 9.10 lower pannel
Only the kappa parameter showed high autocorrelation => typical of higher-level parameters that control the variance of lower-level parameters.
The chains for K show good overlap. The marginal posterior on K spans a wide range of values, and the estimates of the other parameters are not hugely affected by small changes in K.
It can be seen that the prior for the therapeutic touch data is uniform on the group-level mode,
=> equal prior credibility to any possible bias.
The priors on the differences of individual biases: triangular distributions, peaked at a difference of zero.
Shrinkage of the estimates in hierarchical models: the estimates of low-level parameters are pulled closer together if there were not a higher-level distribution.
In Figure 9.10, the most credible values of individual-level biases,
$\therefore$ The variance between the estimated values
Shrinkage causes low-level parameters to shift toward the modes of the higher-level distribution.
Concept at the book's blog
Examples at the book's blog and in Kruschke and Vanpaemel (in press).
Shrunken parameter estimates are less affected by random sampling noise than estimates derived without hierarchical structure.
Shrinkage occurs because the estimate ofeach low-level parameter is influenced from two sources:
(1) the subset ofdata that are directly dependent on the low-level parameter
(2) the higher-level parameters on which the low-level parameter depends.
It is important to understand that shrinkage is a consequence of hierarchical model structure, not Bayesian estimation.
Maximum likelihood estimate (MLE) for the hierarchical model of Figure 9.7
For the whole set of data,
Methods to speed up processing ofJAGS.
- changes the likelihood function in the JAGS model specification. - runjags package to run chains in parallel on multicore computers.
Use the binomial likelihood function instead of the Bernoulli likelihood.
=> remove constant called binomial codefficient,
#z = aggregate( y , by=list(s) , FUN=sum )$x
#N = aggregate( rep(1,length(y)) , by=list(s) , FUN=sum )$x
#Nsubj = length(unique(s))
# Specify the data in a list, for later shipment to JAGS:
#dataList = list(
# z = z ,
# N = N ,
# Nsubj = Nsubj
#)
#model {
# for ( s in 1:Nsubj ) {
# z[s] ~ dbin( theta[s] , N[s] )
# theta[s] ~ dbeta( omega*(kappa-2)+1 , (1-omega)*(kappa-2)+1 )
# }
# omega ~ dbeta( 1 , 1 )
# kappa <- kappaMinusTwo + 2
#kappaMinusTwo ~ dgamma( 0.01 , 0.01 ) # mean=1 , sd=10 (generic vague)
# kappaMinusTwo ~ dgamma( 1.105125 , 0.1051249 ) # mode=1 , sd=10
#}
#source("Jags-Ydich-XnomSsubj-MbinomBetaOmegaKappa.R")Data with many dichonomous value, different categries
# Read the data
myData = read.csv("BattingAverage.csv")
# Load the relevant model into R's working memory:
source("Jags-Ybinom-XnomSsubjCcat-MbinomBetaOmegaKappa.R")##
## *********************************************************************
## Kruschke, J. K. (2015). Doing Bayesian Data Analysis, Second Edition:
## A Tutorial with R, JAGS, and Stan. Academic Press / Elsevier.
## *********************************************************************
# Generate the MCMC chain => Takes few minutes
mcmcCoda = genMCMC(data=myData,
zName="Hits", NName="AtBats", sName="Player",
cName="PriPos", numSavedSteps=11000, thinSteps=20 )## Calling 3 simulations using the parallel method...
## Following the progress of chain 1 (the program will wait for all
## chains to finish before continuing):
## Welcome to JAGS 3.4.0 on Thu Apr 30 16:59:26 2015
## JAGS is free software and comes with ABSOLUTELY NO WARRANTY
## Loading module: basemod: ok
## Loading module: bugs: ok
## . . Reading data file data.txt
## . Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph Size: 3887
## . Reading parameter file inits1.txt
## . Initializing model
## . Adapting 500
## -------------------------------------------------| 500
## ++++++++++++++++++++++++++++++++++++++++++++++++++ 100%
## Adaptation successful
## . Updating 500
## -------------------------------------------------| 500
## ************************************************** 100%
## . . . . . . Updating 73340
## -------------------------------------------------| 73300
## ************************************************** 100%
## * 100%
## . . . . Updating 0
## . Deleting model
## .
## All chains have finished
## Simulation complete. Reading coda files...
## Coda files loaded successfully
## Finished running the simulation
# Display diagnostics of chain, for specified parameters:
for ( parName in c("omega[1]","omegaO","kappa[1]","kappaO", "theta[1]")) {
diagMCMC( codaObject=mcmcCoda, parName=parName,
#saveName=fileNameRoot, saveType=graphFileType #Error in diagMCMC(codaObject = mcmcCoda, parName = parName, saveName = fileNameRoot, : object 'fileNameRoot' not found
#saveName=".", saveType=graphFileType #Error in paste0(file, ".", type) : object 'graphFileType' not found
)
}
Output 9.4
Output 9.5
Output 9.6
Output 9.7
Output 9.8
# Get summary statistics of chain:
summaryInfo = smryMCMC(mcmcCoda, compVal=NULL)## Mean Median Mode ESS HDImass
## omega[1] 0.25159326 0.2516630 0.25138561 11001.0 0.95
## omega[2] 0.24850260 0.2486090 0.24826605 11001.0 0.95
## omega[3] 0.25507674 0.2551890 0.25483945 11001.0 0.95
## omega[4] 0.23704318 0.2371150 0.23720053 10661.0 0.95
## omega[5] 0.25652220 0.2565810 0.25637517 11001.0 0.95
## omega[6] 0.24865038 0.2487270 0.24800075 11001.0 0.95
## omega[7] 0.12213498 0.1222280 0.12187346 4671.5 0.95
## omega[8] 0.25893812 0.2590540 0.25939081 11001.0 0.95
## omega[9] 0.24679522 0.2469180 0.24727568 11001.0 0.95
## omegaO 0.22677181 0.2276580 0.22823408 11001.0 0.95
## kappa[1] 239.60694337 229.6020000 209.76149000 10551.7 0.95
## kappa[2] 258.36181192 245.7180000 235.34277275 11001.0 0.95
## kappa[3] 212.11760569 201.1700000 189.59474997 10662.2 0.95
## kappa[4] 171.83327603 165.7040000 156.63959371 11001.0 0.95
## kappa[5] 240.54239324 228.9570000 209.88914877 11001.0 0.95
## kappa[6] 216.58705853 206.9070000 202.35099549 10579.9 0.95
## kappa[7] 108.07017949 96.6090000 80.51339792 2286.8 0.95
## kappa[8] 403.99603245 381.8180000 351.07285839 11001.0 0.95
## kappa[9] 244.23345480 230.4370000 204.33784384 10240.0 0.95
## kappaO 61.72947343 56.4081000 45.63762885 10259.7 0.95
## theta[1] 0.13129015 0.1288460 0.12389872 11001.0 0.95
## theta[2] 0.24659862 0.2461820 0.24436321 10675.5 0.95
## theta[3] 0.25243199 0.2520200 0.25228599 11749.1 0.95
## theta[4] 0.23325839 0.2330580 0.23170779 10996.3 0.95
## theta[5] 0.25087375 0.2504760 0.25161985 11001.0 0.95
## theta[6] 0.12885334 0.1261530 0.12444015 11001.0 0.95
## theta[7] 0.12908216 0.1265310 0.11948342 11855.8 0.95
## theta[8] 0.24130004 0.2413320 0.24450839 10469.5 0.95
## theta[9] 0.26739032 0.2671580 0.26578383 10805.0 0.95
## theta[10] 0.27791961 0.2777170 0.27788814 11001.0 0.95
## theta[11] 0.12871685 0.1258990 0.12311658 11001.0 0.95
## theta[12] 0.24781849 0.2475180 0.24708975 11001.0 0.95
## theta[13] 0.24497144 0.2445350 0.24440618 11349.6 0.95
## theta[14] 0.24101133 0.2398870 0.23764437 10691.3 0.95
## theta[15] 0.24625511 0.2459880 0.24694220 11667.6 0.95
## theta[16] 0.22657484 0.2264040 0.22526061 11001.0 0.95
## theta[17] 0.27595090 0.2755900 0.27393410 11001.0 0.95
## theta[18] 0.24605900 0.2458150 0.24400501 11001.0 0.95
## theta[19] 0.27653617 0.2763050 0.27744571 11001.0 0.95
## theta[20] 0.12606114 0.1240760 0.11953142 11001.0 0.95
## theta[21] 0.23548199 0.2349900 0.23383802 11001.0 0.95
## theta[22] 0.26518314 0.2651480 0.26527256 11001.0 0.95
## theta[23] 0.12693923 0.1246520 0.11955343 11001.0 0.95
## theta[24] 0.13575087 0.1341040 0.12770993 11001.0 0.95
## theta[25] 0.24210000 0.2416310 0.23994907 11001.0 0.95
## theta[26] 0.24210875 0.2418210 0.24194923 12395.4 0.95
## theta[27] 0.13693193 0.1344050 0.13246828 11001.0 0.95
## theta[28] 0.24943127 0.2492550 0.24771304 11001.0 0.95
## theta[29] 0.12903246 0.1265500 0.12062228 10597.9 0.95
## theta[30] 0.27742498 0.2770670 0.27398078 12730.4 0.95
## theta[31] 0.12941909 0.1266910 0.12011078 11001.0 0.95
## theta[32] 0.12381296 0.1222740 0.11926135 11001.0 0.95
## theta[33] 0.25371029 0.2535880 0.25335699 10590.5 0.95
## theta[34] 0.23904523 0.2384470 0.23375204 11001.0 0.95
## theta[35] 0.12903533 0.1266160 0.12373580 11001.0 0.95
## theta[36] 0.21770369 0.2173700 0.21688516 12335.0 0.95
## theta[37] 0.12728353 0.1251440 0.12459961 11001.0 0.95
## theta[38] 0.23567130 0.2353380 0.23189619 11001.0 0.95
## theta[39] 0.24244320 0.2422700 0.24154385 11001.0 0.95
## theta[40] 0.25270884 0.2527250 0.25420829 11001.0 0.95
## theta[41] 0.21752699 0.2179570 0.21834522 11502.1 0.95
## theta[42] 0.23689247 0.2366850 0.23911775 11001.0 0.95
## theta[43] 0.12997293 0.1281610 0.11991255 11001.0 0.95
## theta[44] 0.12273738 0.1207560 0.12075418 8781.6 0.95
## theta[45] 0.12459292 0.1222070 0.11834389 11001.0 0.95
## theta[46] 0.25133067 0.2513580 0.25157682 10857.3 0.95
## theta[47] 0.26092224 0.2607230 0.26047840 11001.0 0.95
## theta[48] 0.20985404 0.2098380 0.21105976 11001.0 0.95
## theta[49] 0.14225170 0.1395710 0.13333966 10546.3 0.95
## theta[50] 0.12802766 0.1256160 0.12073400 10589.1 0.95
## theta[51] 0.11489591 0.1132490 0.11017330 10164.6 0.95
## theta[52] 0.23940052 0.2392690 0.23807618 11001.0 0.95
## theta[53] 0.12423599 0.1226390 0.11453549 11001.0 0.95
## theta[54] 0.12615119 0.1240560 0.12786913 11001.0 0.95
## theta[55] 0.24012928 0.2400920 0.24130100 10825.5 0.95
## theta[56] 0.12866151 0.1257060 0.12040268 11001.0 0.95
## theta[57] 0.26717828 0.2669250 0.26880963 11001.0 0.95
## theta[58] 0.26513819 0.2647770 0.26374671 11001.0 0.95
## theta[59] 0.30491638 0.3045990 0.30200403 11001.0 0.95
## theta[60] 0.12869461 0.1268010 0.12870721 11001.0 0.95
## theta[61] 0.12416296 0.1218530 0.11782028 10052.5 0.95
## theta[62] 0.25522342 0.2545860 0.25289639 11001.0 0.95
## theta[63] 0.27203869 0.2716360 0.26847780 11001.0 0.95
## theta[64] 0.25517482 0.2549250 0.25550791 11001.0 0.95
## theta[65] 0.24021211 0.2400890 0.23787580 11001.0 0.95
## theta[66] 0.25955443 0.2593050 0.25902566 11001.0 0.95
## theta[67] 0.23465850 0.2347200 0.23333505 11001.0 0.95
## theta[68] 0.14870000 0.1462520 0.14465268 10700.2 0.95
## theta[69] 0.24760012 0.2478080 0.24828020 12791.3 0.95
## theta[70] 0.12708964 0.1246580 0.12246230 11997.3 0.95
## theta[71] 0.12710169 0.1244750 0.11520167 11001.0 0.95
## theta[72] 0.26569077 0.2653760 0.26463868 11382.5 0.95
## theta[73] 0.25197708 0.2516320 0.25152876 11001.0 0.95
## theta[74] 0.22565844 0.2247530 0.21833935 11001.0 0.95
## theta[75] 0.25249391 0.2516100 0.25085847 11001.0 0.95
## theta[76] 0.10952058 0.1082060 0.10229689 9568.0 0.95
## theta[77] 0.28027496 0.2798630 0.28020380 11001.0 0.95
## theta[78] 0.24323747 0.2428460 0.24220928 11001.0 0.95
## theta[79] 0.24949079 0.2494640 0.25010180 10656.3 0.95
## theta[80] 0.23272971 0.2327880 0.23268216 11001.0 0.95
## theta[81] 0.25854101 0.2582640 0.25441851 10635.7 0.95
## theta[82] 0.23781766 0.2366930 0.23081534 11001.0 0.95
## theta[83] 0.25824882 0.2579830 0.26316232 11001.0 0.95
## theta[84] 0.24207379 0.2419920 0.24011677 11001.0 0.95
## theta[85] 0.26969738 0.2695860 0.27113199 11001.0 0.95
## theta[86] 0.12720997 0.1253200 0.11733171 10393.1 0.95
## theta[87] 0.12881839 0.1260050 0.11862584 9991.8 0.95
## theta[88] 0.27919363 0.2790210 0.27939712 11001.0 0.95
## theta[89] 0.25940649 0.2586950 0.26149140 11476.1 0.95
## theta[90] 0.30155400 0.3012080 0.29884968 11001.0 0.95
## theta[91] 0.12573311 0.1232110 0.11993857 11001.0 0.95
## theta[92] 0.23575488 0.2357410 0.22912327 11001.0 0.95
## theta[93] 0.12784476 0.1253270 0.11926034 11001.0 0.95
## theta[94] 0.25376370 0.2535240 0.24834289 11001.0 0.95
## theta[95] 0.24484656 0.2441630 0.24166567 11001.0 0.95
## theta[96] 0.25216416 0.2518700 0.24864307 11001.0 0.95
## theta[97] 0.12951898 0.1270150 0.12506404 11001.0 0.95
## theta[98] 0.25520461 0.2550780 0.25628499 11001.0 0.95
## theta[99] 0.12760118 0.1253600 0.12273710 11001.0 0.95
## theta[100] 0.20799087 0.2075380 0.20682075 11001.0 0.95
## theta[101] 0.24244318 0.2421090 0.24271207 11001.0 0.95
## theta[102] 0.09534844 0.0943767 0.09216475 8194.8 0.95
## theta[103] 0.14370372 0.1424350 0.13669018 11001.0 0.95
## theta[104] 0.10384066 0.1026480 0.09993205 9505.5 0.95
## theta[105] 0.12870244 0.1261430 0.11878623 11001.0 0.95
## theta[106] 0.23700406 0.2366100 0.23623549 10642.8 0.95
## theta[107] 0.24649646 0.2460530 0.24034019 11836.3 0.95
## theta[108] 0.22323678 0.2228460 0.22141687 11001.0 0.95
## theta[109] 0.29643397 0.2960790 0.29544760 11001.0 0.95
## theta[110] 0.23959965 0.2395600 0.24154780 11313.9 0.95
## theta[111] 0.26415143 0.2638570 0.26394430 10634.2 0.95
## theta[112] 0.12859876 0.1258230 0.12138346 11001.0 0.95
## theta[113] 0.24733823 0.2470690 0.24565573 11001.0 0.95
## theta[114] 0.31660029 0.3163570 0.31823272 10012.6 0.95
## theta[115] 0.31156635 0.3111870 0.30892972 11001.0 0.95
## theta[116] 0.12578248 0.1243660 0.11882931 11001.0 0.95
## theta[117] 0.26238284 0.2619280 0.26052179 11104.3 0.95
## theta[118] 0.14949631 0.1473320 0.14348156 11001.0 0.95
## theta[119] 0.22733396 0.2271700 0.22939647 11576.9 0.95
## theta[120] 0.25499048 0.2548390 0.25266605 11001.0 0.95
## theta[121] 0.25386110 0.2534100 0.25258561 11001.0 0.95
## theta[122] 0.26137654 0.2608490 0.25936412 11001.0 0.95
## theta[123] 0.29473574 0.2942820 0.29271451 11001.0 0.95
## theta[124] 0.25680722 0.2560320 0.25432195 11001.0 0.95
## theta[125] 0.11707489 0.1156170 0.11441745 10680.6 0.95
## theta[126] 0.23705003 0.2367780 0.23854925 11862.9 0.95
## theta[127] 0.23845662 0.2378130 0.23523680 11001.0 0.95
## theta[128] 0.23496859 0.2346390 0.23066793 11454.9 0.95
## theta[129] 0.13221528 0.1302410 0.12862414 11001.0 0.95
## theta[130] 0.13918563 0.1361350 0.13070549 10534.8 0.95
## theta[131] 0.27632195 0.2760360 0.27651037 10862.4 0.95
## theta[132] 0.12890677 0.1264270 0.12035412 10624.7 0.95
## theta[133] 0.26007444 0.2597620 0.26085455 11001.0 0.95
## theta[134] 0.25588143 0.2548740 0.25120005 10614.0 0.95
## theta[135] 0.26201497 0.2617120 0.26297415 10775.7 0.95
## theta[136] 0.24533223 0.2450220 0.24542335 10321.9 0.95
## theta[137] 0.24657734 0.2465200 0.24337956 11001.0 0.95
## theta[138] 0.13137186 0.1292040 0.12724102 11001.0 0.95
## theta[139] 0.24518924 0.2449340 0.24472093 11001.0 0.95
## theta[140] 0.13649586 0.1331250 0.12731788 11001.0 0.95
## theta[141] 0.24307333 0.2426280 0.24265185 11529.3 0.95
## theta[142] 0.25256895 0.2520010 0.25027600 11423.0 0.95
## theta[143] 0.25043592 0.2502660 0.25388130 11001.0 0.95
## theta[144] 0.27409754 0.2739260 0.27729552 11001.0 0.95
## theta[145] 0.12781301 0.1256530 0.12280810 11001.0 0.95
## theta[146] 0.25413532 0.2537280 0.25408428 11001.0 0.95
## theta[147] 0.12912204 0.1267790 0.12688222 11001.0 0.95
## theta[148] 0.23875071 0.2376280 0.23065028 11546.5 0.95
## theta[149] 0.27935499 0.2790320 0.27693813 11001.0 0.95
## theta[150] 0.16293044 0.1592490 0.15169599 9904.3 0.95
## theta[151] 0.25166373 0.2516380 0.25280442 11001.0 0.95
## theta[152] 0.14704092 0.1441790 0.13985289 10128.8 0.95
## theta[153] 0.23037171 0.2301400 0.23027562 11356.2 0.95
## theta[154] 0.27097015 0.2706180 0.27121995 11001.0 0.95
## theta[155] 0.12803825 0.1261370 0.12106783 11001.0 0.95
## theta[156] 0.13406798 0.1315370 0.12775261 11083.3 0.95
## theta[157] 0.12662386 0.1250320 0.12626792 10601.6 0.95
## theta[158] 0.26218211 0.2617720 0.26405575 12015.8 0.95
## theta[159] 0.27381066 0.2734510 0.27023736 11001.0 0.95
## theta[160] 0.22392776 0.2237990 0.22024294 11001.0 0.95
## theta[161] 0.12901676 0.1265890 0.11938034 11001.0 0.95
## theta[162] 0.27787141 0.2772890 0.27602186 10663.9 0.95
## theta[163] 0.12897020 0.1265990 0.12298680 11001.0 0.95
## theta[164] 0.24373408 0.2432050 0.24264854 11346.0 0.95
## theta[165] 0.12852712 0.1258990 0.12254929 11001.0 0.95
## theta[166] 0.21856327 0.2183580 0.21592110 11001.0 0.95
## theta[167] 0.12616331 0.1246620 0.12549981 11001.0 0.95
## theta[168] 0.21622513 0.2162860 0.21685058 11001.0 0.95
## theta[169] 0.13692652 0.1343620 0.13124339 11001.0 0.95
## theta[170] 0.12971784 0.1273170 0.12528277 10683.0 0.95
## theta[171] 0.12110878 0.1195020 0.11929805 11001.0 0.95
## theta[172] 0.12508973 0.1223490 0.11544275 11001.0 0.95
## theta[173] 0.12900438 0.1264520 0.12320117 11652.9 0.95
## theta[174] 0.27634092 0.2759680 0.27648355 11001.0 0.95
## theta[175] 0.23316823 0.2321960 0.22987346 11001.0 0.95
## theta[176] 0.21604905 0.2157720 0.21245351 11001.0 0.95
## theta[177] 0.25091952 0.2505290 0.25041168 10668.4 0.95
## theta[178] 0.27122811 0.2704770 0.26723489 11001.0 0.95
## theta[179] 0.12539389 0.1235650 0.11815638 11001.0 0.95
## theta[180] 0.12852735 0.1258670 0.12398880 11001.0 0.95
## theta[181] 0.24978972 0.2491410 0.25217912 11456.4 0.95
## theta[182] 0.12231352 0.1210920 0.11836662 10123.3 0.95
## theta[183] 0.23891313 0.2385410 0.23912209 11001.0 0.95
## theta[184] 0.23215470 0.2321170 0.22698100 11001.0 0.95
## theta[185] 0.25726490 0.2564780 0.25513892 11001.0 0.95
## theta[186] 0.24720273 0.2469390 0.24671037 11001.0 0.95
## theta[187] 0.28935120 0.2892670 0.29048640 11001.0 0.95
## theta[188] 0.24875580 0.2485440 0.24803105 11001.0 0.95
## theta[189] 0.26245876 0.2619150 0.26308785 11001.0 0.95
## theta[190] 0.25921864 0.2591080 0.26113382 10667.4 0.95
## theta[191] 0.28054687 0.2801490 0.27813284 11001.0 0.95
## theta[192] 0.25982532 0.2596630 0.26043195 11001.0 0.95
## theta[193] 0.12869759 0.1265460 0.12329487 10622.6 0.95
## theta[194] 0.24634513 0.2457920 0.24400888 11001.0 0.95
## theta[195] 0.25990679 0.2596100 0.25720369 11001.0 0.95
## theta[196] 0.11381102 0.1125520 0.11359979 10238.6 0.95
## theta[197] 0.24106620 0.2412470 0.24517273 11001.0 0.95
## theta[198] 0.22682356 0.2267790 0.22929209 9516.2 0.95
## theta[199] 0.24442279 0.2442550 0.24793735 10512.4 0.95
## theta[200] 0.23696160 0.2367700 0.23629304 11325.4 0.95
## theta[201] 0.12638707 0.1240300 0.11879836 11001.0 0.95
## theta[202] 0.24456756 0.2443880 0.24446475 11001.0 0.95
## theta[203] 0.24999852 0.2494460 0.24504442 11001.0 0.95
## theta[204] 0.13695840 0.1337630 0.13024647 11001.0 0.95
## theta[205] 0.26504656 0.2648340 0.26620937 11001.0 0.95
## theta[206] 0.23563367 0.2354220 0.23538700 11001.0 0.95
## theta[207] 0.25558680 0.2555040 0.25583432 11001.0 0.95
## theta[208] 0.13809988 0.1352320 0.12997416 10829.6 0.95
## theta[209] 0.27368954 0.2735070 0.27051558 11001.0 0.95
## theta[210] 0.24613783 0.2456920 0.24286122 11001.0 0.95
## theta[211] 0.14457995 0.1410710 0.13348785 11001.0 0.95
## theta[212] 0.26173202 0.2616000 0.25986337 11001.0 0.95
## theta[213] 0.15297532 0.1502110 0.14646460 11001.0 0.95
## theta[214] 0.12130023 0.1197610 0.11566042 11001.0 0.95
## theta[215] 0.27601624 0.2755970 0.27377280 11001.0 0.95
## theta[216] 0.23691386 0.2367520 0.23374470 11364.3 0.95
## theta[217] 0.23675648 0.2366240 0.23517350 11001.0 0.95
## theta[218] 0.27877071 0.2783070 0.27489321 11001.0 0.95
## theta[219] 0.10311301 0.1018120 0.09848354 8126.4 0.95
## theta[220] 0.23838378 0.2383180 0.23808631 11001.0 0.95
## theta[221] 0.13067342 0.1284620 0.12353745 11001.0 0.95
## theta[222] 0.25040183 0.2501010 0.24736973 10172.3 0.95
## theta[223] 0.24139297 0.2410490 0.24211128 11001.0 0.95
## theta[224] 0.25393772 0.2531660 0.25091307 11001.0 0.95
## theta[225] 0.14004447 0.1382720 0.13829969 11001.0 0.95
## theta[226] 0.13808354 0.1349650 0.13052830 11001.0 0.95
## theta[227] 0.29357469 0.2931740 0.29514673 11001.0 0.95
## theta[228] 0.27425730 0.2742320 0.27637815 11001.0 0.95
## theta[229] 0.26719018 0.2665300 0.26606037 11001.0 0.95
## theta[230] 0.23635764 0.2361470 0.23852093 11001.0 0.95
## theta[231] 0.24762067 0.2476260 0.24789757 11320.4 0.95
## theta[232] 0.12864559 0.1265030 0.12302049 11001.0 0.95
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## theta[776] 0.21424670 0.2143230 0.21531678 11001.0 0.95
## theta[777] 0.12374642 0.1212870 0.11285152 11001.0 0.95
## theta[778] 0.12927022 0.1267310 0.11826716 11001.0 0.95
## theta[779] 0.12801976 0.1252050 0.12110698 11001.0 0.95
## theta[780] 0.22728064 0.2270610 0.22982464 11001.0 0.95
## theta[781] 0.11993282 0.1190780 0.11825655 11001.0 0.95
## theta[782] 0.12915516 0.1271670 0.12040381 11001.0 0.95
## theta[783] 0.12301572 0.1212550 0.12001132 9013.6 0.95
## theta[784] 0.23308247 0.2326320 0.23051997 11001.0 0.95
## theta[785] 0.26314835 0.2626120 0.26040163 11001.0 0.95
## theta[786] 0.12415539 0.1222590 0.12124212 11001.0 0.95
## theta[787] 0.27415451 0.2737170 0.27506506 11001.0 0.95
## theta[788] 0.24327184 0.2431210 0.24824620 11001.0 0.95
## theta[789] 0.24956379 0.2492750 0.24582851 11001.0 0.95
## theta[790] 0.12517097 0.1227790 0.11617848 11001.0 0.95
## theta[791] 0.11829962 0.1170610 0.11609647 11001.0 0.95
## theta[792] 0.22935638 0.2292970 0.23121816 11001.0 0.95
## theta[793] 0.23455534 0.2337690 0.23264936 11001.0 0.95
## theta[794] 0.26268826 0.2623100 0.26308586 11451.5 0.95
## theta[795] 0.11882504 0.1170550 0.11583428 11001.0 0.95
## theta[796] 0.25054584 0.2504150 0.25030586 11001.0 0.95
## theta[797] 0.12697658 0.1245780 0.11809332 11001.0 0.95
## theta[798] 0.12897167 0.1264980 0.12018212 11001.0 0.95
## theta[799] 0.23141196 0.2314290 0.23209770 10973.3 0.95
## theta[800] 0.26299655 0.2624930 0.26218712 11001.0 0.95
## theta[801] 0.12797959 0.1253670 0.12263000 11001.0 0.95
## theta[802] 0.25906251 0.2591140 0.25986647 11001.0 0.95
## theta[803] 0.23005211 0.2295450 0.23194852 11503.3 0.95
## theta[804] 0.23518992 0.2347830 0.23487674 11001.0 0.95
## theta[805] 0.26370366 0.2631740 0.26084431 11001.0 0.95
## theta[806] 0.13574989 0.1329780 0.12804039 11001.0 0.95
## theta[807] 0.23983713 0.2399530 0.24285456 10009.9 0.95
## theta[808] 0.29029236 0.2897990 0.28716433 11001.0 0.95
## theta[809] 0.25880639 0.2585780 0.25731931 10963.4 0.95
## theta[810] 0.25279484 0.2520790 0.25057241 11232.2 0.95
## theta[811] 0.24119211 0.2407190 0.23847148 11001.0 0.95
## theta[812] 0.11459069 0.1130690 0.11226277 10361.2 0.95
## theta[813] 0.13316596 0.1301870 0.12439575 11001.0 0.95
## theta[814] 0.23633095 0.2361290 0.23847262 11001.0 0.95
## theta[815] 0.25002959 0.2499280 0.24950871 10069.3 0.95
## theta[816] 0.26607892 0.2652170 0.26056110 11001.0 0.95
## theta[817] 0.12478203 0.1224200 0.11973525 11001.0 0.95
## theta[818] 0.12089797 0.1191100 0.11962851 11227.3 0.95
## theta[819] 0.24428532 0.2438580 0.23898388 10219.4 0.95
## theta[820] 0.22938988 0.2291490 0.22812608 10698.5 0.95
## theta[821] 0.12845383 0.1263660 0.12545054 11001.0 0.95
## theta[822] 0.25721030 0.2572300 0.25927449 11001.0 0.95
## theta[823] 0.25924294 0.2587440 0.25984540 11001.0 0.95
## theta[824] 0.20390627 0.2035460 0.20225257 11001.0 0.95
## theta[825] 0.22341360 0.2237100 0.22885785 11001.0 0.95
## theta[826] 0.27434160 0.2738320 0.27194612 11001.0 0.95
## theta[827] 0.25349485 0.2524530 0.25237574 11001.0 0.95
## theta[828] 0.23447769 0.2341920 0.23788358 11001.0 0.95
## theta[829] 0.25929018 0.2590890 0.25661047 11001.0 0.95
## theta[830] 0.21233082 0.2120120 0.21022150 11001.0 0.95
## theta[831] 0.27542130 0.2749420 0.27320084 11359.8 0.95
## theta[832] 0.24581906 0.2454620 0.25060457 11001.0 0.95
## theta[833] 0.12197304 0.1198610 0.11280610 11363.2 0.95
## theta[834] 0.27592185 0.2755830 0.27752338 11001.0 0.95
## theta[835] 0.12905605 0.1264610 0.12269423 12647.6 0.95
## theta[836] 0.24096971 0.2405600 0.24191273 11001.0 0.95
## theta[837] 0.23163268 0.2311970 0.22560265 11001.0 0.95
## theta[838] 0.12858118 0.1260240 0.12376364 11001.0 0.95
## theta[839] 0.17875310 0.1753800 0.16603003 8645.5 0.95
## theta[840] 0.22761676 0.2273660 0.22638068 11895.8 0.95
## theta[841] 0.15770212 0.1549080 0.14856433 10259.3 0.95
## theta[842] 0.12727946 0.1251140 0.12074554 11001.0 0.95
## theta[843] 0.25227674 0.2517400 0.25141684 11711.6 0.95
## theta[844] 0.27429846 0.2741870 0.27489252 11001.0 0.95
## theta[845] 0.23669665 0.2362840 0.23604881 10465.4 0.95
## theta[846] 0.12801454 0.1254420 0.12315032 11001.0 0.95
## theta[847] 0.26010803 0.2598840 0.25952029 11001.0 0.95
## theta[848] 0.26717333 0.2670660 0.26870244 11001.0 0.95
## theta[849] 0.25250867 0.2523650 0.25103720 11001.0 0.95
## theta[850] 0.25407153 0.2539720 0.25113249 11305.2 0.95
## theta[851] 0.21928443 0.2192180 0.22196376 10589.3 0.95
## theta[852] 0.12919662 0.1261690 0.11896699 10630.5 0.95
## theta[853] 0.25169628 0.2514640 0.25077277 11001.0 0.95
## theta[854] 0.27535892 0.2751950 0.27515441 11001.0 0.95
## theta[855] 0.24265484 0.2421530 0.24026106 10446.3 0.95
## theta[856] 0.24071663 0.2402670 0.23892657 10641.1 0.95
## theta[857] 0.26190630 0.2616780 0.26289019 11001.0 0.95
## theta[858] 0.23620064 0.2358720 0.23383178 11001.0 0.95
## theta[859] 0.25205807 0.2516890 0.24928790 11001.0 0.95
## theta[860] 0.25306220 0.2528140 0.25440360 11348.0 0.95
## theta[861] 0.13362565 0.1307870 0.12726865 11392.6 0.95
## theta[862] 0.23808669 0.2381930 0.23807447 11001.0 0.95
## theta[863] 0.14450100 0.1414040 0.13197567 10466.4 0.95
## theta[864] 0.23267125 0.2322060 0.23329907 11436.1 0.95
## theta[865] 0.24084119 0.2406900 0.24001628 11001.0 0.95
## theta[866] 0.13797957 0.1355860 0.13178811 11351.7 0.95
## theta[867] 0.25785260 0.2571980 0.25306169 11001.0 0.95
## theta[868] 0.21470234 0.2147060 0.21766386 10287.0 0.95
## theta[869] 0.24704957 0.2468220 0.24785022 11001.0 0.95
## theta[870] 0.30586881 0.3053970 0.30400555 11001.0 0.95
## theta[871] 0.26348025 0.2630100 0.26092210 11001.0 0.95
## theta[872] 0.26617823 0.2653890 0.25993582 11431.7 0.95
## theta[873] 0.11311568 0.1117900 0.11125175 10323.1 0.95
## theta[874] 0.25815576 0.2578650 0.25543008 11001.0 0.95
## theta[875] 0.22988243 0.2296980 0.23025919 11001.0 0.95
## theta[876] 0.24969368 0.2496200 0.24958947 10699.1 0.95
## theta[877] 0.27189404 0.2717260 0.27146209 11001.0 0.95
## theta[878] 0.22849189 0.2284450 0.22990938 11001.0 0.95
## theta[879] 0.25372221 0.2534250 0.25102528 11001.0 0.95
## theta[880] 0.23576497 0.2353820 0.23256731 10318.9 0.95
## theta[881] 0.24679256 0.2464690 0.24849390 11001.0 0.95
## theta[882] 0.22945511 0.2290420 0.22707414 11001.0 0.95
## theta[883] 0.22720066 0.2270480 0.22618999 10595.8 0.95
## theta[884] 0.24827480 0.2482950 0.24743262 11001.0 0.95
## theta[885] 0.14386676 0.1403630 0.13438312 11001.0 0.95
## theta[886] 0.25199601 0.2519320 0.25655251 11001.0 0.95
## theta[887] 0.26223460 0.2620480 0.26066558 11001.0 0.95
## theta[888] 0.12498158 0.1230650 0.12037654 10243.0 0.95
## theta[889] 0.25435744 0.2540690 0.25203166 10198.0 0.95
## theta[890] 0.25630987 0.2560990 0.25822685 10287.1 0.95
## theta[891] 0.13769432 0.1349070 0.13262945 11001.0 0.95
## theta[892] 0.12961103 0.1269250 0.12159709 10611.4 0.95
## theta[893] 0.21249489 0.2126640 0.21322887 11001.0 0.95
## theta[894] 0.24464435 0.2443630 0.24428349 11001.0 0.95
## theta[895] 0.11690438 0.1156230 0.11553798 11001.0 0.95
## theta[896] 0.20798556 0.2076350 0.20533788 11001.0 0.95
## theta[897] 0.10863643 0.1075260 0.10712780 9456.7 0.95
## theta[898] 0.14154181 0.1390320 0.13267091 11001.0 0.95
## theta[899] 0.30495537 0.3044520 0.30453243 11001.0 0.95
## theta[900] 0.12663634 0.1252360 0.12288584 11001.0 0.95
## theta[901] 0.26952588 0.2692530 0.26872667 11286.9 0.95
## theta[902] 0.25362353 0.2532850 0.25339186 11753.5 0.95
## theta[903] 0.13298285 0.1307600 0.12893659 11001.0 0.95
## theta[904] 0.12902849 0.1263440 0.12618088 11001.0 0.95
## theta[905] 0.13797572 0.1349760 0.13248668 11001.0 0.95
## theta[906] 0.12941238 0.1265920 0.12006721 11001.0 0.95
## theta[907] 0.23139010 0.2311890 0.22597563 11553.7 0.95
## theta[908] 0.23578609 0.2355870 0.23550514 11001.0 0.95
## theta[909] 0.12345098 0.1211900 0.11879950 11001.0 0.95
## theta[910] 0.23808164 0.2375760 0.23510165 11001.0 0.95
## theta[911] 0.12465114 0.1232220 0.12491596 12632.0 0.95
## theta[912] 0.13548242 0.1325020 0.12610717 9656.3 0.95
## theta[913] 0.24014164 0.2396500 0.23752859 10214.3 0.95
## theta[914] 0.12506457 0.1230910 0.12203932 9788.1 0.95
## theta[915] 0.27788264 0.2772340 0.27491487 11641.1 0.95
## theta[916] 0.12754664 0.1263930 0.12615095 10564.5 0.95
## theta[917] 0.25105315 0.2506230 0.24759694 11870.1 0.95
## theta[918] 0.14189288 0.1395270 0.13381819 11001.0 0.95
## theta[919] 0.23053657 0.2302870 0.23169767 11001.0 0.95
## theta[920] 0.13928556 0.1365160 0.13564390 11001.0 0.95
## theta[921] 0.24704294 0.2468910 0.24732871 11001.0 0.95
## theta[922] 0.24329779 0.2430380 0.24088634 11001.0 0.95
## theta[923] 0.13510064 0.1324020 0.12243578 11001.0 0.95
## theta[924] 0.25754718 0.2574100 0.25465465 11001.0 0.95
## theta[925] 0.22505094 0.2250090 0.22517310 11001.0 0.95
## theta[926] 0.12370455 0.1217240 0.11820357 10568.3 0.95
## theta[927] 0.22997958 0.2302230 0.23251974 10146.7 0.95
## theta[928] 0.25516589 0.2548010 0.25455149 11001.0 0.95
## theta[929] 0.13587478 0.1333390 0.12992637 10636.0 0.95
## theta[930] 0.12907236 0.1267920 0.13002211 10300.4 0.95
## theta[931] 0.15093778 0.1487320 0.14428738 10327.9 0.95
## theta[932] 0.11656853 0.1150940 0.11254639 10333.5 0.95
## theta[933] 0.24272058 0.2425300 0.24076468 11001.0 0.95
## theta[934] 0.29368126 0.2935430 0.29326500 11001.0 0.95
## theta[935] 0.12781495 0.1257490 0.12504650 11001.0 0.95
## theta[936] 0.12802068 0.1257640 0.12239298 11309.4 0.95
## theta[937] 0.24221738 0.2421150 0.24425441 11001.0 0.95
## theta[938] 0.24224326 0.2419580 0.23793723 11001.0 0.95
## theta[939] 0.26225113 0.2620370 0.26068792 11678.3 0.95
## theta[940] 0.28373479 0.2832750 0.28305920 10502.0 0.95
## theta[941] 0.24394953 0.2436120 0.24398754 11663.2 0.95
## theta[942] 0.27054541 0.2704760 0.27117618 11001.0 0.95
## theta[943] 0.14233874 0.1398500 0.13700886 11001.0 0.95
## theta[944] 0.13485981 0.1323610 0.12892028 11001.0 0.95
## theta[945] 0.27584416 0.2755080 0.27496154 11001.0 0.95
## theta[946] 0.15423757 0.1520430 0.14505084 10113.7 0.95
## theta[947] 0.11102572 0.1096300 0.10529834 10203.4 0.95
## theta[948] 0.26565552 0.2652500 0.26324140 11001.0 0.95
## HDIlow HDIhigh CompVal PcntGtCompVal ROPElow ROPEhigh
## omega[1] 0.2423080 0.261246 NA NA NA NA
## omega[2] 0.2377870 0.258159 NA NA NA NA
## omega[3] 0.2443620 0.265510 NA NA NA NA
## omega[4] 0.2268700 0.247580 NA NA NA NA
## omega[5] 0.2453520 0.266944 NA NA NA NA
## omega[6] 0.2388480 0.258123 NA NA NA NA
## omega[7] 0.1098860 0.135106 NA NA NA NA
## omega[8] 0.2496260 0.268180 NA NA NA NA
## omega[9] 0.2357170 0.257678 NA NA NA NA
## omegaO 0.1810690 0.271535 NA NA NA NA
## kappa[1] 124.3330000 379.799000 NA NA NA NA
## kappa[2] 119.9240000 421.676000 NA NA NA NA
## kappa[3] 103.1610000 350.970000 NA NA NA NA
## kappa[4] 89.7164000 264.895000 NA NA NA NA
## kappa[5] 108.2750000 392.293000 NA NA NA NA
## kappa[6] 108.6320000 343.097000 NA NA NA NA
## kappa[7] 41.2895000 201.359000 NA NA NA NA
## kappa[8] 186.3780000 685.258000 NA NA NA NA
## kappa[9] 104.4390000 413.015000 NA NA NA NA
## kappaO 8.8675200 123.316000 NA NA NA NA
## theta[1] 0.0664946 0.197509 NA NA NA NA
## theta[2] 0.2071220 0.288456 NA NA NA NA
## theta[3] 0.2045370 0.302292 NA NA NA NA
## theta[4] 0.2045760 0.261218 NA NA NA NA
## theta[5] 0.2048040 0.301727 NA NA NA NA
## theta[6] 0.0621954 0.198042 NA NA NA NA
## theta[7] 0.0667393 0.202708 NA NA NA NA
## theta[8] 0.1863930 0.297988 NA NA NA NA
## theta[9] 0.2362390 0.298938 NA NA NA NA
## theta[10] 0.2490290 0.310152 NA NA NA NA
## theta[11] 0.0658133 0.201342 NA NA NA NA
## theta[12] 0.2164020 0.279161 NA NA NA NA
## theta[13] 0.2080700 0.282640 NA NA NA NA
## theta[14] 0.1795810 0.308793 NA NA NA NA
## theta[15] 0.1862140 0.304098 NA NA NA NA
## theta[16] 0.1945960 0.260920 NA NA NA NA
## theta[17] 0.2462010 0.306244 NA NA NA NA
## theta[18] 0.2037650 0.291100 NA NA NA NA
## theta[19] 0.2484230 0.307641 NA NA NA NA
## theta[20] 0.0617183 0.194971 NA NA NA NA
## theta[21] 0.1996560 0.273457 NA NA NA NA
## theta[22] 0.2271720 0.303294 NA NA NA NA
## theta[23] 0.0613392 0.194044 NA NA NA NA
## theta[24] 0.0838580 0.188913 NA NA NA NA
## theta[25] 0.1959790 0.292852 NA NA NA NA
## theta[26] 0.2039080 0.276671 NA NA NA NA
## theta[27] 0.0687292 0.207893 NA NA NA NA
## theta[28] 0.2183560 0.280604 NA NA NA NA
## theta[29] 0.0639993 0.198029 NA NA NA NA
## theta[30] 0.2453940 0.309071 NA NA NA NA
## theta[31] 0.0646467 0.202692 NA NA NA NA
## theta[32] 0.0737257 0.173725 NA NA NA NA
## theta[33] 0.2158310 0.289134 NA NA NA NA
## theta[34] 0.1948290 0.283474 NA NA NA NA
## theta[35] 0.0643456 0.199061 NA NA NA NA
## theta[36] 0.1809070 0.254677 NA NA NA NA
## theta[37] 0.0594750 0.194654 NA NA NA NA
## theta[38] 0.2038990 0.267383 NA NA NA NA
## theta[39] 0.1959140 0.291312 NA NA NA NA
## theta[40] 0.2234180 0.283566 NA NA NA NA
## theta[41] 0.1667510 0.265174 NA NA NA NA
## theta[42] 0.1924620 0.284287 NA NA NA NA
## theta[43] 0.0728837 0.189832 NA NA NA NA
## theta[44] 0.0588296 0.187518 NA NA NA NA
## theta[45] 0.0598170 0.190518 NA NA NA NA
## theta[46] 0.2193690 0.283512 NA NA NA NA
## theta[47] 0.2240700 0.297333 NA NA NA NA
## theta[48] 0.1686130 0.251440 NA NA NA NA
## theta[49] 0.0810906 0.207317 NA NA NA NA
## theta[50] 0.0646926 0.199199 NA NA NA NA
## theta[51] 0.0555429 0.172410 NA NA NA NA
## theta[52] 0.2092820 0.270128 NA NA NA NA
## theta[53] 0.0681314 0.180591 NA NA NA NA
## theta[54] 0.0615621 0.192534 NA NA NA NA
## theta[55] 0.1846710 0.294138 NA NA NA NA
## theta[56] 0.0619912 0.198340 NA NA NA NA
## theta[57] 0.2334540 0.302366 NA NA NA NA
## theta[58] 0.2375150 0.294652 NA NA NA NA
## theta[59] 0.2726770 0.337613 NA NA NA NA
## theta[60] 0.0636570 0.198717 NA NA NA NA
## theta[61] 0.0616596 0.193852 NA NA NA NA
## theta[62] 0.2069680 0.304779 NA NA NA NA
## theta[63] 0.2285560 0.313159 NA NA NA NA
## theta[64] 0.2185620 0.293875 NA NA NA NA
## theta[65] 0.2007670 0.279517 NA NA NA NA
## theta[66] 0.2224950 0.295100 NA NA NA NA
## theta[67] 0.1836590 0.283772 NA NA NA NA
## theta[68] 0.0930996 0.210923 NA NA NA NA
## theta[69] 0.2062350 0.288680 NA NA NA NA
## theta[70] 0.0629595 0.196307 NA NA NA NA
## theta[71] 0.0628746 0.198461 NA NA NA NA
## theta[72] 0.2264440 0.306132 NA NA NA NA
## theta[73] 0.2210440 0.282943 NA NA NA NA
## theta[74] 0.1721920 0.283427 NA NA NA NA
## theta[75] 0.1958640 0.311111 NA NA NA NA
## theta[76] 0.0605240 0.161380 NA NA NA NA
## theta[77] 0.2409330 0.317027 NA NA NA NA
## theta[78] 0.1859830 0.300801 NA NA NA NA
## theta[79] 0.2184330 0.277664 NA NA NA NA
## theta[80] 0.2004580 0.264293 NA NA NA NA
## theta[81] 0.2186930 0.299096 NA NA NA NA
## theta[82] 0.1744010 0.303641 NA NA NA NA
## theta[83] 0.2069260 0.309016 NA NA NA NA
## theta[84] 0.1976500 0.284507 NA NA NA NA
## theta[85] 0.2397960 0.298874 NA NA NA NA
## theta[86] 0.0630182 0.196063 NA NA NA NA
## theta[87] 0.0602336 0.197227 NA NA NA NA
## theta[88] 0.2485480 0.311472 NA NA NA NA
## theta[89] 0.2040630 0.309801 NA NA NA NA
## theta[90] 0.2698200 0.334612 NA NA NA NA
## theta[91] 0.0626989 0.194896 NA NA NA NA
## theta[92] 0.1793950 0.289674 NA NA NA NA
## theta[93] 0.0666891 0.199595 NA NA NA NA
## theta[94] 0.2127070 0.292595 NA NA NA NA
## theta[95] 0.1895150 0.302266 NA NA NA NA
## theta[96] 0.2152720 0.287654 NA NA NA NA
## theta[97] 0.0628564 0.197512 NA NA NA NA
## theta[98] 0.2274200 0.283471 NA NA NA NA
## theta[99] 0.0637550 0.195916 NA NA NA NA
## theta[100] 0.1741930 0.244681 NA NA NA NA
## theta[101] 0.1907430 0.294551 NA NA NA NA
## theta[102] 0.0495437 0.141571 NA NA NA NA
## theta[103] 0.0931067 0.199580 NA NA NA NA
## theta[104] 0.0554726 0.151418 NA NA NA NA
## theta[105] 0.0612269 0.197588 NA NA NA NA
## theta[106] 0.1942150 0.279736 NA NA NA NA
## theta[107] 0.1864060 0.306545 NA NA NA NA
## theta[108] 0.1734660 0.273269 NA NA NA NA
## theta[109] 0.2650210 0.327839 NA NA NA NA
## theta[110] 0.1960800 0.284556 NA NA NA NA
## theta[111] 0.2338390 0.295429 NA NA NA NA
## theta[112] 0.0606496 0.195673 NA NA NA NA
## theta[113] 0.2140450 0.281682 NA NA NA NA
## theta[114] 0.2795580 0.353698 NA NA NA NA
## theta[115] 0.2799470 0.344140 NA NA NA NA
## theta[116] 0.0762739 0.175684 NA NA NA NA
## theta[117] 0.2203680 0.303305 NA NA NA NA
## theta[118] 0.0984535 0.207570 NA NA NA NA
## theta[119] 0.1851140 0.270755 NA NA NA NA
## theta[120] 0.2108610 0.300405 NA NA NA NA
## theta[121] 0.2209180 0.287101 NA NA NA NA
## theta[122] 0.2195200 0.306048 NA NA NA NA
## theta[123] 0.2636710 0.326638 NA NA NA NA
## theta[124] 0.2092840 0.310013 NA NA NA NA
## theta[125] 0.0669438 0.169555 NA NA NA NA
## theta[126] 0.1870230 0.285355 NA NA NA NA
## theta[127] 0.1754390 0.301935 NA NA NA NA
## theta[128] 0.1905680 0.279566 NA NA NA NA
## theta[129] 0.0636815 0.197774 NA NA NA NA
## theta[130] 0.0726216 0.210759 NA NA NA NA
## theta[131] 0.2394950 0.316796 NA NA NA NA
## theta[132] 0.0599298 0.194973 NA NA NA NA
## theta[133] 0.2151560 0.306116 NA NA NA NA
## theta[134] 0.1964180 0.312996 NA NA NA NA
## theta[135] 0.2306010 0.294615 NA NA NA NA
## theta[136] 0.1948400 0.297399 NA NA NA NA
## theta[137] 0.2077770 0.287349 NA NA NA NA
## theta[138] 0.0677526 0.198491 NA NA NA NA
## theta[139] 0.2104140 0.283353 NA NA NA NA
## theta[140] 0.0686551 0.204239 NA NA NA NA
## theta[141] 0.1861810 0.300144 NA NA NA NA
## theta[142] 0.2041640 0.295539 NA NA NA NA
## theta[143] 0.2100710 0.292455 NA NA NA NA
## theta[144] 0.2449950 0.304241 NA NA NA NA
## theta[145] 0.0627273 0.196857 NA NA NA NA
## theta[146] 0.2138730 0.297768 NA NA NA NA
## theta[147] 0.0623588 0.198433 NA NA NA NA
## theta[148] 0.1726020 0.306215 NA NA NA NA
## theta[149] 0.2449510 0.311662 NA NA NA NA
## theta[150] 0.0972565 0.236246 NA NA NA NA
## theta[151] 0.1993680 0.302403 NA NA NA NA
## theta[152] 0.0832074 0.217277 NA NA NA NA
## theta[153] 0.1860010 0.274918 NA NA NA NA
## theta[154] 0.2320610 0.312349 NA NA NA NA
## theta[155] 0.0634682 0.197991 NA NA NA NA
## theta[156] 0.0675129 0.202581 NA NA NA NA
## theta[157] 0.0607178 0.194331 NA NA NA NA
## theta[158] 0.2141940 0.307795 NA NA NA NA
## theta[159] 0.2464890 0.302693 NA NA NA NA
## theta[160] 0.1718490 0.277403 NA NA NA NA
## theta[161] 0.0645219 0.199315 NA NA NA NA
## theta[162] 0.2358500 0.318570 NA NA NA NA
## theta[163] 0.0642462 0.199520 NA NA NA NA
## theta[164] 0.1880200 0.298379 NA NA NA NA
## theta[165] 0.0623692 0.196385 NA NA NA NA
## theta[166] 0.1752990 0.262135 NA NA NA NA
## theta[167] 0.0631220 0.189518 NA NA NA NA
## theta[168] 0.1667310 0.264276 NA NA NA NA
## theta[169] 0.0693087 0.210116 NA NA NA NA
## theta[170] 0.0638931 0.196764 NA NA NA NA
## theta[171] 0.0622887 0.185262 NA NA NA NA
## theta[172] 0.0620199 0.192877 NA NA NA NA
## theta[173] 0.0639807 0.199928 NA NA NA NA
## theta[174] 0.2438040 0.307242 NA NA NA NA
## theta[175] 0.1710160 0.296472 NA NA NA NA
## theta[176] 0.1647460 0.264372 NA NA NA NA
## theta[177] 0.1999060 0.303186 NA NA NA NA
## theta[178] 0.2267100 0.318032 NA NA NA NA
## theta[179] 0.0695728 0.183501 NA NA NA NA
## theta[180] 0.0629977 0.199727 NA NA NA NA
## theta[181] 0.1953780 0.305365 NA NA NA NA
## theta[182] 0.0698209 0.176982 NA NA NA NA
## theta[183] 0.1855780 0.294858 NA NA NA NA
## theta[184] 0.1841680 0.282169 NA NA NA NA
## theta[185] 0.2098620 0.308177 NA NA NA NA
## theta[186] 0.2168430 0.276688 NA NA NA NA
## theta[187] 0.2568800 0.324370 NA NA NA NA
## theta[188] 0.2168050 0.281712 NA NA NA NA
## theta[189] 0.2135420 0.311416 NA NA NA NA
## theta[190] 0.2266770 0.291876 NA NA NA NA
## theta[191] 0.2408140 0.321585 NA NA NA NA
## theta[192] 0.2332140 0.288890 NA NA NA NA
## theta[193] 0.0612566 0.197118 NA NA NA NA
## theta[194] 0.1957860 0.295204 NA NA NA NA
## theta[195] 0.2272860 0.291127 NA NA NA NA
## theta[196] 0.0631026 0.162043 NA NA NA NA
## theta[197] 0.1883340 0.295762 NA NA NA NA
## theta[198] 0.1778930 0.274970 NA NA NA NA
## theta[199] 0.1880480 0.299120 NA NA NA NA
## theta[200] 0.1975050 0.279044 NA NA NA NA
## theta[201] 0.0609241 0.192200 NA NA NA NA
## theta[202] 0.1931120 0.296552 NA NA NA NA
## theta[203] 0.1933430 0.309061 NA NA NA NA
## theta[204] 0.0724498 0.209975 NA NA NA NA
## theta[205] 0.2333520 0.297683 NA NA NA NA
## theta[206] 0.2048480 0.266299 NA NA NA NA
## theta[207] 0.2219540 0.289182 NA NA NA NA
## theta[208] 0.0718651 0.210397 NA NA NA NA
## theta[209] 0.2430720 0.307679 NA NA NA NA
## theta[210] 0.1951130 0.297928 NA NA NA NA
## theta[211] 0.0802069 0.220948 NA NA NA NA
## theta[212] 0.2330240 0.290414 NA NA NA NA
## theta[213] 0.0881738 0.218984 NA NA NA NA
## theta[214] 0.0646113 0.177627 NA NA NA NA
## theta[215] 0.2418730 0.307921 NA NA NA NA
## theta[216] 0.1866560 0.287559 NA NA NA NA
## theta[217] 0.2034060 0.269937 NA NA NA NA
## theta[218] 0.2473840 0.312834 NA NA NA NA
## theta[219] 0.0531636 0.155074 NA NA NA NA
## theta[220] 0.1857450 0.289532 NA NA NA NA
## theta[221] 0.0662273 0.195016 NA NA NA NA
## theta[222] 0.1953310 0.306962 NA NA NA NA
## theta[223] 0.1962020 0.290962 NA NA NA NA
## theta[224] 0.1938070 0.313328 NA NA NA NA
## theta[225] 0.0888253 0.192818 NA NA NA NA
## theta[226] 0.0710666 0.211649 NA NA NA NA
## theta[227] 0.2538590 0.333060 NA NA NA NA
## theta[228] 0.2363870 0.312471 NA NA NA NA
## theta[229] 0.2187920 0.317425 NA NA NA NA
## theta[230] 0.1856940 0.280259 NA NA NA NA
## theta[231] 0.2088410 0.285084 NA NA NA NA
## theta[232] 0.0655371 0.201716 NA NA NA NA
## theta[233] 0.2322270 0.300128 NA NA NA NA
## theta[234] 0.1899950 0.272487 NA NA NA NA
## theta[235] 0.2043030 0.276964 NA NA NA NA
## theta[236] 0.0606637 0.194151 NA NA NA NA
## theta[237] 0.1984940 0.271829 NA NA NA NA
## theta[238] 0.2186470 0.280664 NA NA NA NA
## theta[239] 0.0636598 0.199021 NA NA NA NA
## theta[240] 0.0636246 0.198231 NA NA NA NA
## theta[241] 0.1843560 0.263546 NA NA NA NA
## theta[242] 0.1902390 0.250110 NA NA NA NA
## theta[243] 0.0636781 0.199310 NA NA NA NA
## theta[244] 0.2222240 0.298719 NA NA NA NA
## theta[245] 0.2095130 0.307431 NA NA NA NA
## theta[246] 0.0611470 0.197318 NA NA NA NA
## theta[247] 0.2255910 0.296595 NA NA NA NA
## theta[248] 0.2218530 0.289102 NA NA NA NA
## theta[249] 0.2260780 0.302541 NA NA NA NA
## theta[250] 0.1861280 0.285848 NA NA NA NA
## theta[251] 0.2430620 0.305144 NA NA NA NA
## theta[252] 0.0655220 0.178419 NA NA NA NA
## theta[253] 0.2493980 0.311330 NA NA NA NA
## theta[254] 0.1951850 0.288236 NA NA NA NA
## theta[255] 0.2216450 0.282077 NA NA NA NA
## theta[256] 0.2196800 0.277887 NA NA NA NA
## theta[257] 0.0695064 0.177266 NA NA NA NA
## theta[258] 0.2454070 0.303319 NA NA NA NA
## theta[259] 0.1618260 0.288156 NA NA NA NA
## theta[260] 0.2247660 0.324038 NA NA NA NA
## theta[261] 0.0626547 0.198494 NA NA NA NA
## theta[262] 0.1915160 0.301649 NA NA NA NA
## theta[263] 0.1780650 0.309686 NA NA NA NA
## theta[264] 0.0692714 0.205061 NA NA NA NA
## theta[265] 0.0613122 0.194395 NA NA NA NA
## theta[266] 0.1912850 0.303887 NA NA NA NA
## theta[267] 0.2650450 0.329624 NA NA NA NA
## theta[268] 0.0639734 0.177553 NA NA NA NA
## theta[269] 0.0759322 0.212508 NA NA NA NA
## theta[270] 0.1779480 0.264438 NA NA NA NA
## theta[271] 0.0626117 0.195030 NA NA NA NA
## theta[272] 0.1947790 0.279612 NA NA NA NA
## theta[273] 0.1841490 0.261591 NA NA NA NA
## theta[274] 0.1944270 0.282215 NA NA NA NA
## theta[275] 0.1805090 0.275678 NA NA NA NA
## theta[276] 0.0643991 0.201159 NA NA NA NA
## theta[277] 0.2144920 0.309319 NA NA NA NA
## theta[278] 0.1822730 0.284268 NA NA NA NA
## theta[279] 0.2269850 0.300613 NA NA NA NA
## theta[280] 0.2525400 0.320585 NA NA NA NA
## theta[281] 0.0607058 0.171137 NA NA NA NA
## theta[282] 0.2163480 0.289803 NA NA NA NA
## theta[283] 0.2039840 0.291868 NA NA NA NA
## theta[284] 0.2170600 0.272583 NA NA NA NA
## theta[285] 0.2520400 0.343233 NA NA NA NA
## theta[286] 0.2335460 0.303508 NA NA NA NA
## theta[287] 0.2276890 0.288045 NA NA NA NA
## theta[288] 0.2502620 0.316770 NA NA NA NA
## theta[289] 0.0630420 0.178188 NA NA NA NA
## theta[290] 0.1934080 0.312400 NA NA NA NA
## theta[291] 0.1852940 0.293088 NA NA NA NA
## theta[292] 0.2195960 0.301640 NA NA NA NA
## theta[293] 0.2266570 0.290925 NA NA NA NA
## theta[294] 0.0570731 0.183552 NA NA NA NA
## theta[295] 0.0906660 0.198103 NA NA NA NA
## theta[296] 0.1996990 0.281185 NA NA NA NA
## theta[297] 0.2012200 0.302255 NA NA NA NA
## theta[298] 0.2236010 0.310802 NA NA NA NA
## theta[299] 0.0621924 0.198318 NA NA NA NA
## theta[300] 0.1048510 0.232750 NA NA NA NA
## theta[301] 0.2051450 0.319416 NA NA NA NA
## theta[302] 0.0621646 0.172174 NA NA NA NA
## theta[303] 0.0573627 0.184782 NA NA NA NA
## theta[304] 0.0763031 0.195268 NA NA NA NA
## theta[305] 0.2401570 0.322860 NA NA NA NA
## theta[306] 0.0622108 0.183018 NA NA NA NA
## theta[307] 0.2226510 0.305377 NA NA NA NA
## theta[308] 0.1966120 0.292194 NA NA NA NA
## theta[309] 0.2049240 0.287040 NA NA NA NA
## theta[310] 0.1886280 0.307580 NA NA NA NA
## theta[311] 0.1957790 0.301089 NA NA NA NA
## theta[312] 0.1892880 0.320126 NA NA NA NA
## theta[313] 0.2441410 0.309511 NA NA NA NA
## theta[314] 0.0639988 0.198559 NA NA NA NA
## theta[315] 0.2203580 0.296427 NA NA NA NA
## theta[316] 0.1910640 0.283592 NA NA NA NA
## theta[317] 0.2261840 0.294492 NA NA NA NA
## theta[318] 0.0698448 0.207021 NA NA NA NA
## theta[319] 0.2131790 0.307283 NA NA NA NA
## theta[320] 0.2560320 0.317260 NA NA NA NA
## theta[321] 0.1946870 0.300318 NA NA NA NA
## theta[322] 0.2020080 0.302003 NA NA NA NA
## theta[323] 0.2551470 0.321019 NA NA NA NA
## theta[324] 0.0675101 0.197964 NA NA NA NA
## theta[325] 0.0677426 0.167051 NA NA NA NA
## theta[326] 0.2022550 0.283144 NA NA NA NA
## theta[327] 0.2541120 0.314800 NA NA NA NA
## theta[328] 0.2010250 0.272976 NA NA NA NA
## theta[329] 0.0780650 0.215272 NA NA NA NA
## theta[330] 0.2120250 0.286667 NA NA NA NA
## theta[331] 0.2003110 0.317215 NA NA NA NA
## theta[332] 0.2238490 0.317001 NA NA NA NA
## theta[333] 0.2104550 0.268731 NA NA NA NA
## theta[334] 0.1910950 0.309096 NA NA NA NA
## theta[335] 0.1890910 0.280284 NA NA NA NA
## theta[336] 0.2024050 0.274311 NA NA NA NA
## theta[337] 0.0633659 0.197453 NA NA NA NA
## theta[338] 0.1976520 0.311424 NA NA NA NA
## theta[339] 0.0920399 0.217375 NA NA NA NA
## theta[340] 0.0615151 0.177795 NA NA NA NA
## theta[341] 0.2147390 0.302874 NA NA NA NA
## theta[342] 0.1876100 0.307609 NA NA NA NA
## theta[343] 0.2114800 0.286871 NA NA NA NA
## theta[344] 0.2062110 0.282762 NA NA NA NA
## theta[345] 0.0620617 0.195157 NA NA NA NA
## theta[346] 0.1975930 0.298277 NA NA NA NA
## theta[347] 0.2249870 0.307459 NA NA NA NA
## theta[348] 0.2243680 0.294853 NA NA NA NA
## theta[349] 0.1931080 0.309967 NA NA NA NA
## theta[350] 0.0889839 0.201646 NA NA NA NA
## theta[351] 0.1079110 0.224530 NA NA NA NA
## theta[352] 0.2471850 0.309354 NA NA NA NA
## theta[353] 0.0601557 0.191114 NA NA NA NA
## theta[354] 0.0647975 0.201329 NA NA NA NA
## theta[355] 0.0631146 0.197871 NA NA NA NA
## theta[356] 0.2231900 0.301487 NA NA NA NA
## theta[357] 0.2108780 0.287645 NA NA NA NA
## theta[358] 0.0450315 0.143775 NA NA NA NA
## theta[359] 0.0671005 0.179441 NA NA NA NA
## theta[360] 0.0595533 0.159363 NA NA NA NA
## theta[361] 0.2124720 0.268939 NA NA NA NA
## theta[362] 0.0596548 0.192465 NA NA NA NA
## theta[363] 0.2361970 0.298868 NA NA NA NA
## theta[364] 0.0848122 0.192157 NA NA NA NA
## theta[365] 0.1783980 0.279058 NA NA NA NA
## theta[366] 0.2045360 0.280416 NA NA NA NA
## theta[367] 0.0600048 0.189072 NA NA NA NA
## theta[368] 0.2339850 0.295327 NA NA NA NA
## theta[369] 0.0972538 0.237746 NA NA NA NA
## theta[370] 0.0594042 0.195944 NA NA NA NA
## theta[371] 0.1760470 0.275670 NA NA NA NA
## theta[372] 0.2480890 0.309795 NA NA NA NA
## theta[373] 0.2071020 0.302806 NA NA NA NA
## theta[374] 0.0664105 0.185741 NA NA NA NA
## theta[375] 0.2238840 0.297069 NA NA NA NA
## theta[376] 0.0639731 0.200244 NA NA NA NA
## theta[377] 0.2072650 0.286224 NA NA NA NA
## theta[378] 0.0630352 0.198437 NA NA NA NA
## theta[379] 0.0631609 0.198934 NA NA NA NA
## theta[380] 0.2139680 0.304072 NA NA NA NA
## theta[381] 0.0693638 0.211208 NA NA NA NA
## theta[382] 0.0689040 0.208141 NA NA NA NA
## theta[383] 0.1880060 0.275111 NA NA NA NA
## theta[384] 0.0577733 0.191388 NA NA NA NA
## theta[385] 0.1938790 0.317730 NA NA NA NA
## theta[386] 0.1850130 0.273435 NA NA NA NA
## theta[387] 0.0618143 0.195000 NA NA NA NA
## theta[388] 0.2098120 0.296238 NA NA NA NA
## theta[389] 0.2146760 0.295737 NA NA NA NA
## theta[390] 0.1592250 0.283398 NA NA NA NA
## theta[391] 0.1784620 0.283016 NA NA NA NA
## theta[392] 0.2383240 0.294356 NA NA NA NA
## theta[393] 0.1800690 0.281390 NA NA NA NA
## theta[394] 0.2571810 0.318957 NA NA NA NA
## theta[395] 0.1716630 0.288688 NA NA NA NA
## theta[396] 0.1962100 0.308194 NA NA NA NA
## theta[397] 0.0635818 0.200026 NA NA NA NA
## theta[398] 0.1868090 0.275678 NA NA NA NA
## theta[399] 0.0701910 0.203537 NA NA NA NA
## theta[400] 0.1920700 0.313677 NA NA NA NA
## theta[401] 0.1793660 0.307707 NA NA NA NA
## theta[402] 0.0590382 0.192345 NA NA NA NA
## theta[403] 0.2535770 0.316237 NA NA NA NA
## theta[404] 0.2097520 0.309112 NA NA NA NA
## theta[405] 0.2071970 0.268998 NA NA NA NA
## theta[406] 0.1989850 0.275325 NA NA NA NA
## theta[407] 0.0698261 0.198377 NA NA NA NA
## theta[408] 0.1883480 0.262652 NA NA NA NA
## theta[409] 0.1028960 0.224439 NA NA NA NA
## theta[410] 0.1885130 0.287486 NA NA NA NA
## theta[411] 0.0638260 0.200811 NA NA NA NA
## theta[412] 0.1876760 0.304138 NA NA NA NA
## theta[413] 0.0622289 0.196579 NA NA NA NA
## theta[414] 0.0684016 0.203615 NA NA NA NA
## theta[415] 0.1519860 0.235772 NA NA NA NA
## theta[416] 0.0619585 0.195052 NA NA NA NA
## theta[417] 0.2597210 0.320995 NA NA NA NA
## theta[418] 0.0649896 0.197479 NA NA NA NA
## theta[419] 0.1971680 0.282443 NA NA NA NA
## theta[420] 0.2099710 0.279057 NA NA NA NA
## theta[421] 0.1692190 0.269267 NA NA NA NA
## theta[422] 0.2374060 0.299194 NA NA NA NA
## theta[423] 0.1971240 0.271533 NA NA NA NA
## theta[424] 0.2094350 0.298541 NA NA NA NA
## theta[425] 0.2027490 0.287432 NA NA NA NA
## theta[426] 0.2174800 0.294964 NA NA NA NA
## theta[427] 0.2554960 0.320061 NA NA NA NA
## theta[428] 0.1850160 0.277394 NA NA NA NA
## theta[429] 0.1074760 0.228468 NA NA NA NA
## theta[430] 0.1896210 0.297006 NA NA NA NA
## theta[431] 0.1968890 0.308301 NA NA NA NA
## theta[432] 0.1802660 0.264967 NA NA NA NA
## theta[433] 0.2233120 0.303877 NA NA NA NA
## theta[434] 0.2547120 0.323842 NA NA NA NA
## theta[435] 0.0578671 0.195650 NA NA NA NA
## theta[436] 0.2159100 0.278162 NA NA NA NA
## theta[437] 0.2699050 0.331005 NA NA NA NA
## theta[438] 0.0617909 0.191944 NA NA NA NA
## theta[439] 0.2413170 0.307539 NA NA NA NA
## theta[440] 0.2120000 0.322453 NA NA NA NA
## theta[441] 0.2084080 0.282125 NA NA NA NA
## theta[442] 0.0690704 0.172037 NA NA NA NA
## theta[443] 0.2030020 0.264729 NA NA NA NA
## theta[444] 0.1935620 0.289874 NA NA NA NA
## theta[445] 0.2319520 0.312490 NA NA NA NA
## theta[446] 0.1837520 0.298975 NA NA NA NA
## theta[447] 0.2496590 0.309029 NA NA NA NA
## theta[448] 0.1823190 0.262893 NA NA NA NA
## theta[449] 0.2421270 0.315413 NA NA NA NA
## theta[450] 0.2353500 0.299787 NA NA NA NA
## theta[451] 0.2195650 0.281689 NA NA NA NA
## theta[452] 0.0634886 0.187414 NA NA NA NA
## theta[453] 0.2017200 0.292097 NA NA NA NA
## theta[454] 0.2011530 0.296561 NA NA NA NA
## theta[455] 0.0709574 0.188754 NA NA NA NA
## theta[456] 0.1862790 0.278599 NA NA NA NA
## theta[457] 0.2041640 0.293501 NA NA NA NA
## theta[458] 0.0730599 0.203319 NA NA NA NA
## theta[459] 0.2018320 0.281306 NA NA NA NA
## theta[460] 0.0772291 0.195412 NA NA NA NA
## theta[461] 0.2524010 0.323574 NA NA NA NA
## theta[462] 0.2452190 0.306661 NA NA NA NA
## theta[463] 0.0748095 0.190112 NA NA NA NA
## theta[464] 0.2151550 0.306307 NA NA NA NA
## theta[465] 0.0631842 0.165682 NA NA NA NA
## theta[466] 0.2644820 0.340151 NA NA NA NA
## theta[467] 0.1014820 0.218296 NA NA NA NA
## theta[468] 0.0648292 0.185935 NA NA NA NA
## theta[469] 0.2263390 0.282300 NA NA NA NA
## theta[470] 0.2261460 0.284358 NA NA NA NA
## theta[471] 0.0640911 0.199579 NA NA NA NA
## theta[472] 0.2106500 0.296934 NA NA NA NA
## theta[473] 0.2529090 0.317223 NA NA NA NA
## theta[474] 0.0612809 0.195671 NA NA NA NA
## theta[475] 0.2062730 0.269819 NA NA NA NA
## theta[476] 0.2087340 0.300049 NA NA NA NA
## theta[477] 0.1824660 0.270743 NA NA NA NA
## theta[478] 0.2194750 0.324682 NA NA NA NA
## theta[479] 0.1965570 0.292794 NA NA NA NA
## theta[480] 0.2216080 0.285636 NA NA NA NA
## theta[481] 0.0617853 0.195184 NA NA NA NA
## theta[482] 0.0618921 0.197281 NA NA NA NA
## theta[483] 0.2207020 0.292735 NA NA NA NA
## theta[484] 0.1994490 0.313979 NA NA NA NA
## theta[485] 0.2171230 0.309580 NA NA NA NA
## theta[486] 0.1696930 0.296091 NA NA NA NA
## theta[487] 0.1916370 0.311777 NA NA NA NA
## theta[488] 0.0639582 0.193253 NA NA NA NA
## theta[489] 0.2029140 0.305521 NA NA NA NA
## theta[490] 0.2346420 0.295971 NA NA NA NA
## theta[491] 0.0963322 0.208235 NA NA NA NA
## theta[492] 0.1561830 0.248702 NA NA NA NA
## theta[493] 0.2348340 0.300731 NA NA NA NA
## theta[494] 0.1264520 0.260696 NA NA NA NA
## theta[495] 0.0616689 0.186159 NA NA NA NA
## theta[496] 0.0622756 0.195674 NA NA NA NA
## theta[497] 0.2290660 0.290507 NA NA NA NA
## theta[498] 0.0865983 0.192779 NA NA NA NA
## theta[499] 0.2324330 0.313811 NA NA NA NA
## theta[500] 0.1832990 0.303761 NA NA NA NA
## theta[501] 0.0623736 0.196891 NA NA NA NA
## theta[502] 0.1767320 0.305425 NA NA NA NA
## theta[503] 0.0598640 0.189516 NA NA NA NA
## theta[504] 0.0776712 0.216179 NA NA NA NA
## theta[505] 0.1879890 0.301055 NA NA NA NA
## theta[506] 0.1982100 0.293790 NA NA NA NA
## theta[507] 0.1864340 0.268082 NA NA NA NA
## theta[508] 0.0601451 0.182401 NA NA NA NA
## theta[509] 0.2137090 0.304816 NA NA NA NA
## theta[510] 0.0643262 0.171250 NA NA NA NA
## theta[511] 0.0713985 0.204690 NA NA NA NA
## theta[512] 0.2173440 0.290058 NA NA NA NA
## theta[513] 0.0609935 0.195765 NA NA NA NA
## theta[514] 0.1864520 0.276360 NA NA NA NA
## theta[515] 0.0648890 0.190693 NA NA NA NA
## theta[516] 0.0690472 0.167934 NA NA NA NA
## theta[517] 0.2311540 0.300463 NA NA NA NA
## theta[518] 0.2172020 0.284213 NA NA NA NA
## theta[519] 0.2385820 0.319065 NA NA NA NA
## theta[520] 0.2115850 0.293723 NA NA NA NA
## theta[521] 0.2016920 0.305822 NA NA NA NA
## theta[522] 0.0639200 0.198755 NA NA NA NA
## theta[523] 0.0623141 0.199625 NA NA NA NA
## theta[524] 0.0744691 0.216704 NA NA NA NA
## theta[525] 0.2111160 0.281895 NA NA NA NA
## theta[526] 0.2520400 0.334693 NA NA NA NA
## theta[527] 0.2332110 0.301992 NA NA NA NA
## theta[528] 0.0558061 0.179030 NA NA NA NA
## theta[529] 0.1980610 0.298048 NA NA NA NA
## theta[530] 0.1913710 0.304371 NA NA NA NA
## theta[531] 0.0813440 0.197484 NA NA NA NA
## theta[532] 0.0560500 0.156017 NA NA NA NA
## theta[533] 0.2181360 0.305385 NA NA NA NA
## theta[534] 0.0563538 0.152512 NA NA NA NA
## theta[535] 0.1917340 0.306066 NA NA NA NA
## theta[536] 0.1880000 0.289976 NA NA NA NA
## theta[537] 0.0631459 0.197489 NA NA NA NA
## theta[538] 0.1640840 0.292313 NA NA NA NA
## theta[539] 0.2126590 0.297327 NA NA NA NA
## theta[540] 0.0686690 0.180998 NA NA NA NA
## theta[541] 0.2485870 0.312807 NA NA NA NA
## theta[542] 0.1027310 0.236827 NA NA NA NA
## theta[543] 0.1900030 0.276575 NA NA NA NA
## theta[544] 0.0628717 0.200575 NA NA NA NA
## theta[545] 0.2106780 0.298363 NA NA NA NA
## theta[546] 0.1897830 0.296049 NA NA NA NA
## theta[547] 0.1860750 0.253579 NA NA NA NA
## theta[548] 0.0615885 0.193500 NA NA NA NA
## theta[549] 0.1991970 0.294751 NA NA NA NA
## theta[550] 0.2101090 0.279365 NA NA NA NA
## theta[551] 0.1644160 0.289545 NA NA NA NA
## theta[552] 0.1804120 0.269259 NA NA NA NA
## theta[553] 0.0662063 0.199811 NA NA NA NA
## theta[554] 0.2203010 0.296087 NA NA NA NA
## theta[555] 0.1949780 0.274235 NA NA NA NA
## theta[556] 0.1849420 0.270937 NA NA NA NA
## theta[557] 0.1698610 0.265664 NA NA NA NA
## theta[558] 0.0602549 0.196402 NA NA NA NA
## theta[559] 0.0637797 0.199848 NA NA NA NA
## theta[560] 0.1863420 0.305459 NA NA NA NA
## theta[561] 0.0689357 0.205917 NA NA NA NA
## theta[562] 0.2677520 0.336447 NA NA NA NA
## theta[563] 0.2053530 0.277476 NA NA NA NA
## theta[564] 0.2155300 0.282403 NA NA NA NA
## theta[565] 0.2171660 0.278731 NA NA NA NA
## theta[566] 0.1986180 0.300727 NA NA NA NA
## theta[567] 0.0798864 0.221144 NA NA NA NA
## theta[568] 0.2098640 0.291493 NA NA NA NA
## theta[569] 0.1995110 0.266422 NA NA NA NA
## theta[570] 0.0631423 0.199821 NA NA NA NA
## theta[571] 0.0614936 0.194604 NA NA NA NA
## theta[572] 0.1868930 0.287847 NA NA NA NA
## theta[573] 0.2762190 0.341187 NA NA NA NA
## theta[574] 0.1891290 0.288313 NA NA NA NA
## theta[575] 0.0807795 0.191350 NA NA NA NA
## theta[576] 0.2074790 0.289654 NA NA NA NA
## theta[577] 0.1962800 0.268767 NA NA NA NA
## theta[578] 0.0605325 0.192283 NA NA NA NA
## theta[579] 0.1949950 0.277571 NA NA NA NA
## theta[580] 0.2055730 0.284500 NA NA NA NA
## theta[581] 0.0607274 0.190373 NA NA NA NA
## theta[582] 0.0732449 0.188540 NA NA NA NA
## theta[583] 0.0615426 0.193626 NA NA NA NA
## theta[584] 0.0598164 0.195022 NA NA NA NA
## theta[585] 0.1898510 0.290748 NA NA NA NA
## theta[586] 0.2026140 0.319936 NA NA NA NA
## theta[587] 0.1811930 0.271303 NA NA NA NA
## theta[588] 0.2344410 0.314683 NA NA NA NA
## theta[589] 0.0612691 0.196340 NA NA NA NA
## theta[590] 0.0629880 0.195355 NA NA NA NA
## theta[591] 0.0922297 0.201711 NA NA NA NA
## theta[592] 0.0823988 0.225519 NA NA NA NA
## theta[593] 0.0593400 0.188862 NA NA NA NA
## theta[594] 0.0696804 0.203770 NA NA NA NA
## theta[595] 0.0612789 0.159606 NA NA NA NA
## theta[596] 0.1906510 0.271574 NA NA NA NA
## theta[597] 0.2618540 0.331064 NA NA NA NA
## theta[598] 0.2238230 0.288230 NA NA NA NA
## theta[599] 0.2384680 0.307399 NA NA NA NA
## theta[600] 0.1734580 0.301355 NA NA NA NA
## theta[601] 0.0606022 0.189300 NA NA NA NA
## theta[602] 0.2161670 0.301877 NA NA NA NA
## theta[603] 0.2127290 0.303034 NA NA NA NA
## theta[604] 0.0691302 0.208443 NA NA NA NA
## theta[605] 0.2341670 0.298370 NA NA NA NA
## theta[606] 0.1798210 0.276591 NA NA NA NA
## theta[607] 0.2312970 0.305230 NA NA NA NA
## theta[608] 0.2102100 0.285525 NA NA NA NA
## theta[609] 0.2327550 0.296364 NA NA NA NA
## theta[610] 0.2022940 0.276452 NA NA NA NA
## theta[611] 0.2409430 0.312561 NA NA NA NA
## theta[612] 0.0633462 0.199944 NA NA NA NA
## theta[613] 0.0540330 0.176814 NA NA NA NA
## theta[614] 0.0698596 0.202501 NA NA NA NA
## theta[615] 0.0632300 0.197643 NA NA NA NA
## theta[616] 0.2330670 0.313438 NA NA NA NA
## theta[617] 0.2139180 0.275206 NA NA NA NA
## theta[618] 0.0710194 0.198887 NA NA NA NA
## theta[619] 0.2455330 0.308242 NA NA NA NA
## theta[620] 0.2533980 0.323105 NA NA NA NA
## theta[621] 0.1926760 0.288747 NA NA NA NA
## theta[622] 0.1770300 0.265064 NA NA NA NA
## theta[623] 0.1958230 0.268248 NA NA NA NA
## theta[624] 0.0627064 0.194813 NA NA NA NA
## theta[625] 0.2095890 0.288196 NA NA NA NA
## theta[626] 0.1993900 0.312694 NA NA NA NA
## theta[627] 0.1788270 0.285017 NA NA NA NA
## theta[628] 0.1941040 0.308438 NA NA NA NA
## theta[629] 0.2026450 0.319501 NA NA NA NA
## theta[630] 0.2468600 0.323012 NA NA NA NA
## theta[631] 0.0668929 0.188882 NA NA NA NA
## theta[632] 0.1664020 0.265104 NA NA NA NA
## theta[633] 0.1047000 0.223108 NA NA NA NA
## theta[634] 0.2182560 0.294190 NA NA NA NA
## theta[635] 0.2096700 0.319970 NA NA NA NA
## theta[636] 0.1839510 0.291710 NA NA NA NA
## theta[637] 0.2090380 0.295969 NA NA NA NA
## theta[638] 0.2005710 0.295158 NA NA NA NA
## theta[639] 0.0773796 0.219990 NA NA NA NA
## theta[640] 0.0876643 0.198639 NA NA NA NA
## theta[641] 0.0681263 0.174759 NA NA NA NA
## theta[642] 0.1746900 0.259223 NA NA NA NA
## theta[643] 0.0655258 0.197356 NA NA NA NA
## theta[644] 0.2131510 0.311956 NA NA NA NA
## theta[645] 0.0698286 0.207930 NA NA NA NA
## theta[646] 0.0600009 0.182996 NA NA NA NA
## theta[647] 0.1920160 0.267461 NA NA NA NA
## theta[648] 0.2024850 0.313494 NA NA NA NA
## theta[649] 0.1853610 0.293485 NA NA NA NA
## theta[650] 0.2108810 0.313809 NA NA NA NA
## theta[651] 0.2023100 0.321208 NA NA NA NA
## theta[652] 0.2531230 0.331443 NA NA NA NA
## theta[653] 0.0601753 0.195836 NA NA NA NA
## theta[654] 0.0671572 0.196123 NA NA NA NA
## theta[655] 0.0645925 0.187581 NA NA NA NA
## theta[656] 0.2082740 0.299250 NA NA NA NA
## theta[657] 0.0635823 0.198544 NA NA NA NA
## theta[658] 0.2592990 0.328808 NA NA NA NA
## theta[659] 0.0608872 0.197430 NA NA NA NA
## theta[660] 0.2496060 0.311328 NA NA NA NA
## theta[661] 0.2065460 0.289562 NA NA NA NA
## theta[662] 0.0589176 0.186544 NA NA NA NA
## theta[663] 0.2031240 0.285089 NA NA NA NA
## theta[664] 0.2317920 0.302608 NA NA NA NA
## theta[665] 0.0646286 0.199417 NA NA NA NA
## theta[666] 0.1890140 0.279465 NA NA NA NA
## theta[667] 0.2050840 0.289908 NA NA NA NA
## theta[668] 0.2180650 0.321999 NA NA NA NA
## theta[669] 0.1876240 0.299450 NA NA NA NA
## theta[670] 0.0640953 0.197419 NA NA NA NA
## theta[671] 0.2018550 0.290469 NA NA NA NA
## theta[672] 0.0598408 0.189634 NA NA NA NA
## theta[673] 0.2480740 0.310693 NA NA NA NA
## theta[674] 0.2052150 0.290890 NA NA NA NA
## theta[675] 0.1900020 0.306570 NA NA NA NA
## theta[676] 0.0696015 0.201914 NA NA NA NA
## theta[677] 0.1935530 0.278600 NA NA NA NA
## theta[678] 0.1852110 0.246499 NA NA NA NA
## theta[679] 0.1917580 0.307351 NA NA NA NA
## theta[680] 0.2287250 0.282640 NA NA NA NA
## theta[681] 0.1944460 0.259232 NA NA NA NA
## theta[682] 0.0629590 0.197208 NA NA NA NA
## theta[683] 0.2140370 0.273887 NA NA NA NA
## theta[684] 0.0642938 0.191160 NA NA NA NA
## theta[685] 0.1980790 0.316296 NA NA NA NA
## theta[686] 0.1983580 0.310181 NA NA NA NA
## theta[687] 0.0586300 0.197621 NA NA NA NA
## theta[688] 0.2373430 0.319965 NA NA NA NA
## theta[689] 0.1824290 0.261053 NA NA NA NA
## theta[690] 0.0593266 0.196146 NA NA NA NA
## theta[691] 0.0693751 0.206127 NA NA NA NA
## theta[692] 0.1957320 0.299360 NA NA NA NA
## theta[693] 0.2422340 0.303873 NA NA NA NA
## theta[694] 0.2004210 0.316120 NA NA NA NA
## theta[695] 0.2509080 0.324182 NA NA NA NA
## theta[696] 0.2339660 0.302367 NA NA NA NA
## theta[697] 0.1929650 0.285876 NA NA NA NA
## theta[698] 0.1728210 0.304569 NA NA NA NA
## theta[699] 0.2079190 0.275575 NA NA NA NA
## theta[700] 0.2321210 0.320130 NA NA NA NA
## theta[701] 0.2198950 0.295668 NA NA NA NA
## theta[702] 0.2064990 0.305632 NA NA NA NA
## theta[703] 0.0934386 0.221618 NA NA NA NA
## theta[704] 0.0653666 0.198584 NA NA NA NA
## theta[705] 0.2777540 0.347861 NA NA NA NA
## theta[706] 0.2564770 0.319000 NA NA NA NA
## theta[707] 0.2080990 0.279293 NA NA NA NA
## theta[708] 0.0624697 0.193966 NA NA NA NA
## theta[709] 0.2029290 0.318338 NA NA NA NA
## theta[710] 0.1908900 0.298926 NA NA NA NA
## theta[711] 0.2467140 0.306568 NA NA NA NA
## theta[712] 0.1975400 0.285124 NA NA NA NA
## theta[713] 0.2193420 0.296079 NA NA NA NA
## theta[714] 0.0628661 0.193881 NA NA NA NA
## theta[715] 0.2046470 0.288084 NA NA NA NA
## theta[716] 0.1882600 0.284010 NA NA NA NA
## theta[717] 0.1736080 0.304832 NA NA NA NA
## theta[718] 0.1743680 0.253171 NA NA NA NA
## theta[719] 0.0727153 0.204947 NA NA NA NA
## theta[720] 0.2319000 0.291238 NA NA NA NA
## theta[721] 0.2578270 0.321606 NA NA NA NA
## theta[722] 0.0634076 0.198069 NA NA NA NA
## theta[723] 0.0617657 0.193649 NA NA NA NA
## theta[724] 0.2259850 0.286247 NA NA NA NA
## theta[725] 0.0653198 0.200157 NA NA NA NA
## theta[726] 0.1942890 0.303709 NA NA NA NA
## theta[727] 0.1952590 0.271128 NA NA NA NA
## theta[728] 0.2044560 0.263944 NA NA NA NA
## theta[729] 0.1604260 0.272899 NA NA NA NA
## theta[730] 0.2214210 0.275883 NA NA NA NA
## theta[731] 0.0635977 0.197525 NA NA NA NA
## theta[732] 0.2148290 0.319137 NA NA NA NA
## theta[733] 0.1970440 0.311551 NA NA NA NA
## theta[734] 0.0631085 0.191667 NA NA NA NA
## theta[735] 0.2508750 0.311872 NA NA NA NA
## theta[736] 0.2457540 0.306204 NA NA NA NA
## theta[737] 0.2017000 0.265038 NA NA NA NA
## theta[738] 0.0611726 0.192703 NA NA NA NA
## theta[739] 0.1974840 0.286228 NA NA NA NA
## theta[740] 0.0654775 0.161717 NA NA NA NA
## theta[741] 0.0608657 0.194283 NA NA NA NA
## theta[742] 0.2596610 0.317643 NA NA NA NA
## theta[743] 0.2113080 0.284886 NA NA NA NA
## theta[744] 0.2359440 0.309414 NA NA NA NA
## theta[745] 0.1861480 0.283905 NA NA NA NA
## theta[746] 0.2091560 0.275719 NA NA NA NA
## theta[747] 0.2124110 0.299840 NA NA NA NA
## theta[748] 0.1932410 0.283980 NA NA NA NA
## theta[749] 0.2355900 0.302488 NA NA NA NA
## theta[750] 0.0724246 0.212687 NA NA NA NA
## theta[751] 0.1753060 0.308399 NA NA NA NA
## theta[752] 0.0608369 0.193990 NA NA NA NA
## theta[753] 0.1923360 0.266929 NA NA NA NA
## theta[754] 0.0587702 0.155032 NA NA NA NA
## theta[755] 0.0618494 0.189235 NA NA NA NA
## theta[756] 0.0717384 0.208490 NA NA NA NA
## theta[757] 0.0750403 0.207746 NA NA NA NA
## theta[758] 0.2119700 0.286978 NA NA NA NA
## theta[759] 0.2214080 0.279018 NA NA NA NA
## theta[760] 0.0712411 0.203874 NA NA NA NA
## theta[761] 0.2041940 0.317886 NA NA NA NA
## theta[762] 0.1983580 0.290129 NA NA NA NA
## theta[763] 0.2264400 0.297361 NA NA NA NA
## theta[764] 0.0727641 0.211671 NA NA NA NA
## theta[765] 0.2351470 0.294353 NA NA NA NA
## theta[766] 0.2013710 0.293198 NA NA NA NA
## theta[767] 0.0605839 0.196437 NA NA NA NA
## theta[768] 0.0621407 0.195587 NA NA NA NA
## theta[769] 0.1753540 0.280072 NA NA NA NA
## theta[770] 0.2072150 0.320492 NA NA NA NA
## theta[771] 0.2493020 0.328738 NA NA NA NA
## theta[772] 0.2608490 0.339152 NA NA NA NA
## theta[773] 0.0706398 0.200802 NA NA NA NA
## theta[774] 0.0644173 0.200233 NA NA NA NA
## theta[775] 0.2259300 0.301712 NA NA NA NA
## theta[776] 0.1803600 0.247391 NA NA NA NA
## theta[777] 0.0583629 0.187360 NA NA NA NA
## theta[778] 0.0654282 0.198893 NA NA NA NA
## theta[779] 0.0624777 0.196535 NA NA NA NA
## theta[780] 0.1931310 0.261369 NA NA NA NA
## theta[781] 0.0705334 0.175012 NA NA NA NA
## theta[782] 0.0629827 0.197887 NA NA NA NA
## theta[783] 0.0675914 0.182141 NA NA NA NA
## theta[784] 0.1973820 0.270412 NA NA NA NA
## theta[785] 0.2200550 0.308432 NA NA NA NA
## theta[786] 0.0620746 0.193746 NA NA NA NA
## theta[787] 0.2394680 0.310227 NA NA NA NA
## theta[788] 0.1880750 0.302142 NA NA NA NA
## theta[789] 0.2172560 0.281984 NA NA NA NA
## theta[790] 0.0615953 0.193610 NA NA NA NA
## theta[791] 0.0643518 0.175052 NA NA NA NA
## theta[792] 0.1893000 0.267531 NA NA NA NA
## theta[793] 0.1683610 0.297873 NA NA NA NA
## theta[794] 0.2218640 0.306088 NA NA NA NA
## theta[795] 0.0650323 0.177611 NA NA NA NA
## theta[796] 0.2190360 0.280963 NA NA NA NA
## theta[797] 0.0609014 0.195136 NA NA NA NA
## theta[798] 0.0657335 0.202745 NA NA NA NA
## theta[799] 0.1952320 0.267853 NA NA NA NA
## theta[800] 0.2056190 0.317637 NA NA NA NA
## theta[801] 0.0628489 0.200331 NA NA NA NA
## theta[802] 0.2236190 0.292242 NA NA NA NA
## theta[803] 0.1702700 0.297865 NA NA NA NA
## theta[804] 0.1836860 0.288491 NA NA NA NA
## theta[805] 0.2249080 0.301693 NA NA NA NA
## theta[806] 0.0720411 0.206740 NA NA NA NA
## theta[807] 0.2026090 0.275855 NA NA NA NA
## theta[808] 0.2602020 0.320777 NA NA NA NA
## theta[809] 0.2285400 0.288582 NA NA NA NA
## theta[810] 0.2090590 0.297407 NA NA NA NA
## theta[811] 0.1881970 0.293139 NA NA NA NA
## theta[812] 0.0542645 0.175804 NA NA NA NA
## theta[813] 0.0690106 0.203393 NA NA NA NA
## theta[814] 0.1945320 0.280074 NA NA NA NA
## theta[815] 0.2103300 0.286532 NA NA NA NA
## theta[816] 0.2244090 0.312618 NA NA NA NA
## theta[817] 0.0624337 0.194690 NA NA NA NA
## theta[818] 0.0617605 0.187408 NA NA NA NA
## theta[819] 0.2096040 0.279789 NA NA NA NA
## theta[820] 0.1995390 0.261936 NA NA NA NA
## theta[821] 0.0621170 0.196724 NA NA NA NA
## theta[822] 0.2172450 0.292783 NA NA NA NA
## theta[823] 0.2082420 0.310297 NA NA NA NA
## theta[824] 0.1647080 0.245844 NA NA NA NA
## theta[825] 0.1760140 0.270160 NA NA NA NA
## theta[826] 0.2365100 0.314000 NA NA NA NA
## theta[827] 0.1963640 0.317978 NA NA NA NA
## theta[828] 0.1706660 0.299865 NA NA NA NA
## theta[829] 0.2292000 0.290503 NA NA NA NA
## theta[830] 0.1775230 0.251080 NA NA NA NA
## theta[831] 0.2436240 0.308399 NA NA NA NA
## theta[832] 0.1860380 0.306261 NA NA NA NA
## theta[833] 0.0575050 0.187608 NA NA NA NA
## theta[834] 0.2463760 0.307239 NA NA NA NA
## theta[835] 0.0636351 0.199464 NA NA NA NA
## theta[836] 0.1927940 0.289876 NA NA NA NA
## theta[837] 0.1901170 0.277422 NA NA NA NA
## theta[838] 0.0600103 0.195098 NA NA NA NA
## theta[839] 0.1128450 0.246922 NA NA NA NA
## theta[840] 0.1964340 0.258778 NA NA NA NA
## theta[841] 0.0943128 0.224071 NA NA NA NA
## theta[842] 0.0650607 0.194449 NA NA NA NA
## theta[843] 0.2055770 0.300731 NA NA NA NA
## theta[844] 0.2471870 0.302610 NA NA NA NA
## theta[845] 0.2021710 0.272563 NA NA NA NA
## theta[846] 0.0626156 0.197092 NA NA NA NA
## theta[847] 0.2244590 0.296359 NA NA NA NA
## theta[848] 0.2384050 0.295292 NA NA NA NA
## theta[849] 0.2205060 0.284957 NA NA NA NA
## theta[850] 0.2102390 0.298447 NA NA NA NA
## theta[851] 0.1629050 0.273785 NA NA NA NA
## theta[852] 0.0642951 0.198712 NA NA NA NA
## theta[853] 0.2199010 0.283969 NA NA NA NA
## theta[854] 0.2418990 0.308751 NA NA NA NA
## theta[855] 0.1843510 0.300607 NA NA NA NA
## theta[856] 0.2045990 0.280005 NA NA NA NA
## theta[857] 0.2271920 0.297325 NA NA NA NA
## theta[858] 0.1999270 0.275390 NA NA NA NA
## theta[859] 0.2071630 0.295566 NA NA NA NA
## theta[860] 0.2104940 0.296893 NA NA NA NA
## theta[861] 0.0713607 0.204796 NA NA NA NA
## theta[862] 0.1858220 0.291984 NA NA NA NA
## theta[863] 0.0776537 0.218567 NA NA NA NA
## theta[864] 0.1907000 0.276407 NA NA NA NA
## theta[865] 0.2064520 0.276519 NA NA NA NA
## theta[866] 0.0737770 0.206607 NA NA NA NA
## theta[867] 0.2110490 0.306164 NA NA NA NA
## theta[868] 0.1659370 0.266516 NA NA NA NA
## theta[869] 0.1899990 0.301531 NA NA NA NA
## theta[870] 0.2727350 0.338981 NA NA NA NA
## theta[871] 0.2328220 0.296438 NA NA NA NA
## theta[872] 0.2226800 0.309485 NA NA NA NA
## theta[873] 0.0555649 0.175359 NA NA NA NA
## theta[874] 0.2136640 0.297456 NA NA NA NA
## theta[875] 0.1997270 0.259730 NA NA NA NA
## theta[876] 0.2202010 0.281047 NA NA NA NA
## theta[877] 0.2425660 0.299939 NA NA NA NA
## theta[878] 0.1832740 0.272558 NA NA NA NA
## theta[879] 0.2182320 0.291488 NA NA NA NA
## theta[880] 0.1967920 0.275089 NA NA NA NA
## theta[881] 0.2050610 0.290399 NA NA NA NA
## theta[882] 0.1878950 0.269179 NA NA NA NA
## theta[883] 0.1814750 0.270345 NA NA NA NA
## theta[884] 0.2050760 0.291534 NA NA NA NA
## theta[885] 0.0750333 0.215635 NA NA NA NA
## theta[886] 0.1974390 0.304887 NA NA NA NA
## theta[887] 0.2320180 0.293085 NA NA NA NA
## theta[888] 0.0587783 0.189994 NA NA NA NA
## theta[889] 0.2213440 0.285671 NA NA NA NA
## theta[890] 0.2278780 0.287585 NA NA NA NA
## theta[891] 0.0713309 0.210041 NA NA NA NA
## theta[892] 0.0606023 0.198257 NA NA NA NA
## theta[893] 0.1600640 0.260540 NA NA NA NA
## theta[894] 0.2034900 0.289024 NA NA NA NA
## theta[895] 0.0682865 0.170954 NA NA NA NA
## theta[896] 0.1475690 0.268120 NA NA NA NA
## theta[897] 0.0594308 0.158858 NA NA NA NA
## theta[898] 0.0802389 0.204393 NA NA NA NA
## theta[899] 0.2677690 0.343256 NA NA NA NA
## theta[900] 0.0769056 0.178340 NA NA NA NA
## theta[901] 0.2373650 0.302054 NA NA NA NA
## theta[902] 0.2128060 0.291485 NA NA NA NA
## theta[903] 0.0661798 0.200829 NA NA NA NA
## theta[904] 0.0649638 0.199728 NA NA NA NA
## theta[905] 0.0719047 0.210471 NA NA NA NA
## theta[906] 0.0656138 0.201127 NA NA NA NA
## theta[907] 0.1991770 0.262587 NA NA NA NA
## theta[908] 0.2075890 0.265652 NA NA NA NA
## theta[909] 0.0578712 0.188671 NA NA NA NA
## theta[910] 0.1998710 0.275916 NA NA NA NA
## theta[911] 0.0651854 0.190432 NA NA NA NA
## theta[912] 0.0697802 0.206950 NA NA NA NA
## theta[913] 0.2017480 0.280926 NA NA NA NA
## theta[914] 0.0651210 0.191402 NA NA NA NA
## theta[915] 0.2436720 0.312430 NA NA NA NA
## theta[916] 0.0749319 0.182184 NA NA NA NA
## theta[917] 0.2024220 0.300116 NA NA NA NA
## theta[918] 0.0814437 0.205092 NA NA NA NA
## theta[919] 0.1703580 0.296488 NA NA NA NA
## theta[920] 0.0751733 0.210324 NA NA NA NA
## theta[921] 0.2148000 0.278238 NA NA NA NA
## theta[922] 0.2072220 0.279907 NA NA NA NA
## theta[923] 0.0655562 0.205046 NA NA NA NA
## theta[924] 0.2267650 0.290246 NA NA NA NA
## theta[925] 0.1787760 0.267523 NA NA NA NA
## theta[926] 0.0594062 0.188603 NA NA NA NA
## theta[927] 0.1795580 0.278284 NA NA NA NA
## theta[928] 0.2147610 0.297553 NA NA NA NA
## theta[929] 0.0814731 0.197481 NA NA NA NA
## theta[930] 0.0669352 0.201959 NA NA NA NA
## theta[931] 0.0950204 0.210141 NA NA NA NA
## theta[932] 0.0613635 0.170901 NA NA NA NA
## theta[933] 0.1947150 0.291160 NA NA NA NA
## theta[934] 0.2619800 0.326168 NA NA NA NA
## theta[935] 0.0591143 0.193840 NA NA NA NA
## theta[936] 0.0627301 0.196901 NA NA NA NA
## theta[937] 0.2086540 0.276066 NA NA NA NA
## theta[938] 0.2066480 0.279116 NA NA NA NA
## theta[939] 0.2322840 0.293600 NA NA NA NA
## theta[940] 0.2361550 0.327019 NA NA NA NA
## theta[941] 0.1826710 0.301701 NA NA NA NA
## theta[942] 0.2405720 0.300649 NA NA NA NA
## theta[943] 0.0840591 0.205019 NA NA NA NA
## theta[944] 0.0676633 0.204236 NA NA NA NA
## theta[945] 0.2453250 0.307839 NA NA NA NA
## theta[946] 0.0996233 0.214938 NA NA NA NA
## theta[947] 0.0606587 0.160782 NA NA NA NA
## theta[948] 0.2362520 0.293203 NA NA NA NA
## PcntLtROPE PcntInROPE PcntGtROPE
## omega[1] NA NA NA
## omega[2] NA NA NA
## omega[3] NA NA NA
## omega[4] NA NA NA
## omega[5] NA NA NA
## omega[6] NA NA NA
## omega[7] NA NA NA
## omega[8] NA NA NA
## omega[9] NA NA NA
## omegaO NA NA NA
## kappa[1] NA NA NA
## kappa[2] NA NA NA
## kappa[3] NA NA NA
## kappa[4] NA NA NA
## kappa[5] NA NA NA
## kappa[6] NA NA NA
## kappa[7] NA NA NA
## kappa[8] NA NA NA
## kappa[9] NA NA NA
## kappaO NA NA NA
## theta[1] NA NA NA
## theta[2] NA NA NA
## theta[3] NA NA NA
## theta[4] NA NA NA
## theta[5] NA NA NA
## theta[6] NA NA NA
## theta[7] NA NA NA
## theta[8] NA NA NA
## theta[9] NA NA NA
## theta[10] NA NA NA
## theta[11] NA NA NA
## theta[12] NA NA NA
## theta[13] NA NA NA
## theta[14] NA NA NA
## theta[15] NA NA NA
## theta[16] NA NA NA
## theta[17] NA NA NA
## theta[18] NA NA NA
## theta[19] NA NA NA
## theta[20] NA NA NA
## theta[21] NA NA NA
## theta[22] NA NA NA
## theta[23] NA NA NA
## theta[24] NA NA NA
## theta[25] NA NA NA
## theta[26] NA NA NA
## theta[27] NA NA NA
## theta[28] NA NA NA
## theta[29] NA NA NA
## theta[30] NA NA NA
## theta[31] NA NA NA
## theta[32] NA NA NA
## theta[33] NA NA NA
## theta[34] NA NA NA
## theta[35] NA NA NA
## theta[36] NA NA NA
## theta[37] NA NA NA
## theta[38] NA NA NA
## theta[39] NA NA NA
## theta[40] NA NA NA
## theta[41] NA NA NA
## theta[42] NA NA NA
## theta[43] NA NA NA
## theta[44] NA NA NA
## theta[45] NA NA NA
## theta[46] NA NA NA
## theta[47] NA NA NA
## theta[48] NA NA NA
## theta[49] NA NA NA
## theta[50] NA NA NA
## theta[51] NA NA NA
## theta[52] NA NA NA
## theta[53] NA NA NA
## theta[54] NA NA NA
## theta[55] NA NA NA
## theta[56] NA NA NA
## theta[57] NA NA NA
## theta[58] NA NA NA
## theta[59] NA NA NA
## theta[60] NA NA NA
## theta[61] NA NA NA
## theta[62] NA NA NA
## theta[63] NA NA NA
## theta[64] NA NA NA
## theta[65] NA NA NA
## theta[66] NA NA NA
## theta[67] NA NA NA
## theta[68] NA NA NA
## theta[69] NA NA NA
## theta[70] NA NA NA
## theta[71] NA NA NA
## theta[72] NA NA NA
## theta[73] NA NA NA
## theta[74] NA NA NA
## theta[75] NA NA NA
## theta[76] NA NA NA
## theta[77] NA NA NA
## theta[78] NA NA NA
## theta[79] NA NA NA
## theta[80] NA NA NA
## theta[81] NA NA NA
## theta[82] NA NA NA
## theta[83] NA NA NA
## theta[84] NA NA NA
## theta[85] NA NA NA
## theta[86] NA NA NA
## theta[87] NA NA NA
## theta[88] NA NA NA
## theta[89] NA NA NA
## theta[90] NA NA NA
## theta[91] NA NA NA
## theta[92] NA NA NA
## theta[93] NA NA NA
## theta[94] NA NA NA
## theta[95] NA NA NA
## theta[96] NA NA NA
## theta[97] NA NA NA
## theta[98] NA NA NA
## theta[99] NA NA NA
## theta[100] NA NA NA
## theta[101] NA NA NA
## theta[102] NA NA NA
## theta[103] NA NA NA
## theta[104] NA NA NA
## theta[105] NA NA NA
## theta[106] NA NA NA
## theta[107] NA NA NA
## theta[108] NA NA NA
## theta[109] NA NA NA
## theta[110] NA NA NA
## theta[111] NA NA NA
## theta[112] NA NA NA
## theta[113] NA NA NA
## theta[114] NA NA NA
## theta[115] NA NA NA
## theta[116] NA NA NA
## theta[117] NA NA NA
## theta[118] NA NA NA
## theta[119] NA NA NA
## theta[120] NA NA NA
## theta[121] NA NA NA
## theta[122] NA NA NA
## theta[123] NA NA NA
## theta[124] NA NA NA
## theta[125] NA NA NA
## theta[126] NA NA NA
## theta[127] NA NA NA
## theta[128] NA NA NA
## theta[129] NA NA NA
## theta[130] NA NA NA
## theta[131] NA NA NA
## theta[132] NA NA NA
## theta[133] NA NA NA
## theta[134] NA NA NA
## theta[135] NA NA NA
## theta[136] NA NA NA
## theta[137] NA NA NA
## theta[138] NA NA NA
## theta[139] NA NA NA
## theta[140] NA NA NA
## theta[141] NA NA NA
## theta[142] NA NA NA
## theta[143] NA NA NA
## theta[144] NA NA NA
## theta[145] NA NA NA
## theta[146] NA NA NA
## theta[147] NA NA NA
## theta[148] NA NA NA
## theta[149] NA NA NA
## theta[150] NA NA NA
## theta[151] NA NA NA
## theta[152] NA NA NA
## theta[153] NA NA NA
## theta[154] NA NA NA
## theta[155] NA NA NA
## theta[156] NA NA NA
## theta[157] NA NA NA
## theta[158] NA NA NA
## theta[159] NA NA NA
## theta[160] NA NA NA
## theta[161] NA NA NA
## theta[162] NA NA NA
## theta[163] NA NA NA
## theta[164] NA NA NA
## theta[165] NA NA NA
## theta[166] NA NA NA
## theta[167] NA NA NA
## theta[168] NA NA NA
## theta[169] NA NA NA
## theta[170] NA NA NA
## theta[171] NA NA NA
## theta[172] NA NA NA
## theta[173] NA NA NA
## theta[174] NA NA NA
## theta[175] NA NA NA
## theta[176] NA NA NA
## theta[177] NA NA NA
## theta[178] NA NA NA
## theta[179] NA NA NA
## theta[180] NA NA NA
## theta[181] NA NA NA
## theta[182] NA NA NA
## theta[183] NA NA NA
## theta[184] NA NA NA
## theta[185] NA NA NA
## theta[186] NA NA NA
## theta[187] NA NA NA
## theta[188] NA NA NA
## theta[189] NA NA NA
## theta[190] NA NA NA
## theta[191] NA NA NA
## theta[192] NA NA NA
## theta[193] NA NA NA
## theta[194] NA NA NA
## theta[195] NA NA NA
## theta[196] NA NA NA
## theta[197] NA NA NA
## theta[198] NA NA NA
## theta[199] NA NA NA
## theta[200] NA NA NA
## theta[201] NA NA NA
## theta[202] NA NA NA
## theta[203] NA NA NA
## theta[204] NA NA NA
## theta[205] NA NA NA
## theta[206] NA NA NA
## theta[207] NA NA NA
## theta[208] NA NA NA
## theta[209] NA NA NA
## theta[210] NA NA NA
## theta[211] NA NA NA
## theta[212] NA NA NA
## theta[213] NA NA NA
## theta[214] NA NA NA
## theta[215] NA NA NA
## theta[216] NA NA NA
## theta[217] NA NA NA
## theta[218] NA NA NA
## theta[219] NA NA NA
## theta[220] NA NA NA
## theta[221] NA NA NA
## theta[222] NA NA NA
## theta[223] NA NA NA
## theta[224] NA NA NA
## theta[225] NA NA NA
## theta[226] NA NA NA
## theta[227] NA NA NA
## theta[228] NA NA NA
## theta[229] NA NA NA
## theta[230] NA NA NA
## theta[231] NA NA NA
## theta[232] NA NA NA
## theta[233] NA NA NA
## theta[234] NA NA NA
## theta[235] NA NA NA
## theta[236] NA NA NA
## theta[237] NA NA NA
## theta[238] NA NA NA
## theta[239] NA NA NA
## theta[240] NA NA NA
## theta[241] NA NA NA
## theta[242] NA NA NA
## theta[243] NA NA NA
## theta[244] NA NA NA
## theta[245] NA NA NA
## theta[246] NA NA NA
## theta[247] NA NA NA
## theta[248] NA NA NA
## theta[249] NA NA NA
## theta[250] NA NA NA
## theta[251] NA NA NA
## theta[252] NA NA NA
## theta[253] NA NA NA
## theta[254] NA NA NA
## theta[255] NA NA NA
## theta[256] NA NA NA
## theta[257] NA NA NA
## theta[258] NA NA NA
## theta[259] NA NA NA
## theta[260] NA NA NA
## theta[261] NA NA NA
## theta[262] NA NA NA
## theta[263] NA NA NA
## theta[264] NA NA NA
## theta[265] NA NA NA
## theta[266] NA NA NA
## theta[267] NA NA NA
## theta[268] NA NA NA
## theta[269] NA NA NA
## theta[270] NA NA NA
## theta[271] NA NA NA
## theta[272] NA NA NA
## theta[273] NA NA NA
## theta[274] NA NA NA
## theta[275] NA NA NA
## theta[276] NA NA NA
## theta[277] NA NA NA
## theta[278] NA NA NA
## theta[279] NA NA NA
## theta[280] NA NA NA
## theta[281] NA NA NA
## theta[282] NA NA NA
## theta[283] NA NA NA
## theta[284] NA NA NA
## theta[285] NA NA NA
## theta[286] NA NA NA
## theta[287] NA NA NA
## theta[288] NA NA NA
## theta[289] NA NA NA
## theta[290] NA NA NA
## theta[291] NA NA NA
## theta[292] NA NA NA
## theta[293] NA NA NA
## theta[294] NA NA NA
## theta[295] NA NA NA
## theta[296] NA NA NA
## theta[297] NA NA NA
## theta[298] NA NA NA
## theta[299] NA NA NA
## theta[300] NA NA NA
## theta[301] NA NA NA
## theta[302] NA NA NA
## theta[303] NA NA NA
## theta[304] NA NA NA
## theta[305] NA NA NA
## theta[306] NA NA NA
## theta[307] NA NA NA
## theta[308] NA NA NA
## theta[309] NA NA NA
## theta[310] NA NA NA
## theta[311] NA NA NA
## theta[312] NA NA NA
## theta[313] NA NA NA
## theta[314] NA NA NA
## theta[315] NA NA NA
## theta[316] NA NA NA
## theta[317] NA NA NA
## theta[318] NA NA NA
## theta[319] NA NA NA
## theta[320] NA NA NA
## theta[321] NA NA NA
## theta[322] NA NA NA
## theta[323] NA NA NA
## theta[324] NA NA NA
## theta[325] NA NA NA
## theta[326] NA NA NA
## theta[327] NA NA NA
## theta[328] NA NA NA
## theta[329] NA NA NA
## theta[330] NA NA NA
## theta[331] NA NA NA
## theta[332] NA NA NA
## theta[333] NA NA NA
## theta[334] NA NA NA
## theta[335] NA NA NA
## theta[336] NA NA NA
## theta[337] NA NA NA
## theta[338] NA NA NA
## theta[339] NA NA NA
## theta[340] NA NA NA
## theta[341] NA NA NA
## theta[342] NA NA NA
## theta[343] NA NA NA
## theta[344] NA NA NA
## theta[345] NA NA NA
## theta[346] NA NA NA
## theta[347] NA NA NA
## theta[348] NA NA NA
## theta[349] NA NA NA
## theta[350] NA NA NA
## theta[351] NA NA NA
## theta[352] NA NA NA
## theta[353] NA NA NA
## theta[354] NA NA NA
## theta[355] NA NA NA
## theta[356] NA NA NA
## theta[357] NA NA NA
## theta[358] NA NA NA
## theta[359] NA NA NA
## theta[360] NA NA NA
## theta[361] NA NA NA
## theta[362] NA NA NA
## theta[363] NA NA NA
## theta[364] NA NA NA
## theta[365] NA NA NA
## theta[366] NA NA NA
## theta[367] NA NA NA
## theta[368] NA NA NA
## theta[369] NA NA NA
## theta[370] NA NA NA
## theta[371] NA NA NA
## theta[372] NA NA NA
## theta[373] NA NA NA
## theta[374] NA NA NA
## theta[375] NA NA NA
## theta[376] NA NA NA
## theta[377] NA NA NA
## theta[378] NA NA NA
## theta[379] NA NA NA
## theta[380] NA NA NA
## theta[381] NA NA NA
## theta[382] NA NA NA
## theta[383] NA NA NA
## theta[384] NA NA NA
## theta[385] NA NA NA
## theta[386] NA NA NA
## theta[387] NA NA NA
## theta[388] NA NA NA
## theta[389] NA NA NA
## theta[390] NA NA NA
## theta[391] NA NA NA
## theta[392] NA NA NA
## theta[393] NA NA NA
## theta[394] NA NA NA
## theta[395] NA NA NA
## theta[396] NA NA NA
## theta[397] NA NA NA
## theta[398] NA NA NA
## theta[399] NA NA NA
## theta[400] NA NA NA
## theta[401] NA NA NA
## theta[402] NA NA NA
## theta[403] NA NA NA
## theta[404] NA NA NA
## theta[405] NA NA NA
## theta[406] NA NA NA
## theta[407] NA NA NA
## theta[408] NA NA NA
## theta[409] NA NA NA
## theta[410] NA NA NA
## theta[411] NA NA NA
## theta[412] NA NA NA
## theta[413] NA NA NA
## theta[414] NA NA NA
## theta[415] NA NA NA
## theta[416] NA NA NA
## theta[417] NA NA NA
## theta[418] NA NA NA
## theta[419] NA NA NA
## theta[420] NA NA NA
## theta[421] NA NA NA
## theta[422] NA NA NA
## theta[423] NA NA NA
## theta[424] NA NA NA
## theta[425] NA NA NA
## theta[426] NA NA NA
## theta[427] NA NA NA
## theta[428] NA NA NA
## theta[429] NA NA NA
## theta[430] NA NA NA
## theta[431] NA NA NA
## theta[432] NA NA NA
## theta[433] NA NA NA
## theta[434] NA NA NA
## theta[435] NA NA NA
## theta[436] NA NA NA
## theta[437] NA NA NA
## theta[438] NA NA NA
## theta[439] NA NA NA
## theta[440] NA NA NA
## theta[441] NA NA NA
## theta[442] NA NA NA
## theta[443] NA NA NA
## theta[444] NA NA NA
## theta[445] NA NA NA
## theta[446] NA NA NA
## theta[447] NA NA NA
## theta[448] NA NA NA
## theta[449] NA NA NA
## theta[450] NA NA NA
## theta[451] NA NA NA
## theta[452] NA NA NA
## theta[453] NA NA NA
## theta[454] NA NA NA
## theta[455] NA NA NA
## theta[456] NA NA NA
## theta[457] NA NA NA
## theta[458] NA NA NA
## theta[459] NA NA NA
## theta[460] NA NA NA
## theta[461] NA NA NA
## theta[462] NA NA NA
## theta[463] NA NA NA
## theta[464] NA NA NA
## theta[465] NA NA NA
## theta[466] NA NA NA
## theta[467] NA NA NA
## theta[468] NA NA NA
## theta[469] NA NA NA
## theta[470] NA NA NA
## theta[471] NA NA NA
## theta[472] NA NA NA
## theta[473] NA NA NA
## theta[474] NA NA NA
## theta[475] NA NA NA
## theta[476] NA NA NA
## theta[477] NA NA NA
## theta[478] NA NA NA
## theta[479] NA NA NA
## theta[480] NA NA NA
## theta[481] NA NA NA
## theta[482] NA NA NA
## theta[483] NA NA NA
## theta[484] NA NA NA
## theta[485] NA NA NA
## theta[486] NA NA NA
## theta[487] NA NA NA
## theta[488] NA NA NA
## theta[489] NA NA NA
## theta[490] NA NA NA
## theta[491] NA NA NA
## theta[492] NA NA NA
## theta[493] NA NA NA
## theta[494] NA NA NA
## theta[495] NA NA NA
## theta[496] NA NA NA
## theta[497] NA NA NA
## theta[498] NA NA NA
## theta[499] NA NA NA
## theta[500] NA NA NA
## theta[501] NA NA NA
## theta[502] NA NA NA
## theta[503] NA NA NA
## theta[504] NA NA NA
## theta[505] NA NA NA
## theta[506] NA NA NA
## theta[507] NA NA NA
## theta[508] NA NA NA
## theta[509] NA NA NA
## theta[510] NA NA NA
## theta[511] NA NA NA
## theta[512] NA NA NA
## theta[513] NA NA NA
## theta[514] NA NA NA
## theta[515] NA NA NA
## theta[516] NA NA NA
## theta[517] NA NA NA
## theta[518] NA NA NA
## theta[519] NA NA NA
## theta[520] NA NA NA
## theta[521] NA NA NA
## theta[522] NA NA NA
## theta[523] NA NA NA
## theta[524] NA NA NA
## theta[525] NA NA NA
## theta[526] NA NA NA
## theta[527] NA NA NA
## theta[528] NA NA NA
## theta[529] NA NA NA
## theta[530] NA NA NA
## theta[531] NA NA NA
## theta[532] NA NA NA
## theta[533] NA NA NA
## theta[534] NA NA NA
## theta[535] NA NA NA
## theta[536] NA NA NA
## theta[537] NA NA NA
## theta[538] NA NA NA
## theta[539] NA NA NA
## theta[540] NA NA NA
## theta[541] NA NA NA
## theta[542] NA NA NA
## theta[543] NA NA NA
## theta[544] NA NA NA
## theta[545] NA NA NA
## theta[546] NA NA NA
## theta[547] NA NA NA
## theta[548] NA NA NA
## theta[549] NA NA NA
## theta[550] NA NA NA
## theta[551] NA NA NA
## theta[552] NA NA NA
## theta[553] NA NA NA
## theta[554] NA NA NA
## theta[555] NA NA NA
## theta[556] NA NA NA
## theta[557] NA NA NA
## theta[558] NA NA NA
## theta[559] NA NA NA
## theta[560] NA NA NA
## theta[561] NA NA NA
## theta[562] NA NA NA
## theta[563] NA NA NA
## theta[564] NA NA NA
## theta[565] NA NA NA
## theta[566] NA NA NA
## theta[567] NA NA NA
## theta[568] NA NA NA
## theta[569] NA NA NA
## theta[570] NA NA NA
## theta[571] NA NA NA
## theta[572] NA NA NA
## theta[573] NA NA NA
## theta[574] NA NA NA
## theta[575] NA NA NA
## theta[576] NA NA NA
## theta[577] NA NA NA
## theta[578] NA NA NA
## theta[579] NA NA NA
## theta[580] NA NA NA
## theta[581] NA NA NA
## theta[582] NA NA NA
## theta[583] NA NA NA
## theta[584] NA NA NA
## theta[585] NA NA NA
## theta[586] NA NA NA
## theta[587] NA NA NA
## theta[588] NA NA NA
## theta[589] NA NA NA
## theta[590] NA NA NA
## theta[591] NA NA NA
## theta[592] NA NA NA
## theta[593] NA NA NA
## theta[594] NA NA NA
## theta[595] NA NA NA
## theta[596] NA NA NA
## theta[597] NA NA NA
## theta[598] NA NA NA
## theta[599] NA NA NA
## theta[600] NA NA NA
## theta[601] NA NA NA
## theta[602] NA NA NA
## theta[603] NA NA NA
## theta[604] NA NA NA
## theta[605] NA NA NA
## theta[606] NA NA NA
## theta[607] NA NA NA
## theta[608] NA NA NA
## theta[609] NA NA NA
## theta[610] NA NA NA
## theta[611] NA NA NA
## theta[612] NA NA NA
## theta[613] NA NA NA
## theta[614] NA NA NA
## theta[615] NA NA NA
## theta[616] NA NA NA
## theta[617] NA NA NA
## theta[618] NA NA NA
## theta[619] NA NA NA
## theta[620] NA NA NA
## theta[621] NA NA NA
## theta[622] NA NA NA
## theta[623] NA NA NA
## theta[624] NA NA NA
## theta[625] NA NA NA
## theta[626] NA NA NA
## theta[627] NA NA NA
## theta[628] NA NA NA
## theta[629] NA NA NA
## theta[630] NA NA NA
## theta[631] NA NA NA
## theta[632] NA NA NA
## theta[633] NA NA NA
## theta[634] NA NA NA
## theta[635] NA NA NA
## theta[636] NA NA NA
## theta[637] NA NA NA
## theta[638] NA NA NA
## theta[639] NA NA NA
## theta[640] NA NA NA
## theta[641] NA NA NA
## theta[642] NA NA NA
## theta[643] NA NA NA
## theta[644] NA NA NA
## theta[645] NA NA NA
## theta[646] NA NA NA
## theta[647] NA NA NA
## theta[648] NA NA NA
## theta[649] NA NA NA
## theta[650] NA NA NA
## theta[651] NA NA NA
## theta[652] NA NA NA
## theta[653] NA NA NA
## theta[654] NA NA NA
## theta[655] NA NA NA
## theta[656] NA NA NA
## theta[657] NA NA NA
## theta[658] NA NA NA
## theta[659] NA NA NA
## theta[660] NA NA NA
## theta[661] NA NA NA
## theta[662] NA NA NA
## theta[663] NA NA NA
## theta[664] NA NA NA
## theta[665] NA NA NA
## theta[666] NA NA NA
## theta[667] NA NA NA
## theta[668] NA NA NA
## theta[669] NA NA NA
## theta[670] NA NA NA
## theta[671] NA NA NA
## theta[672] NA NA NA
## theta[673] NA NA NA
## theta[674] NA NA NA
## theta[675] NA NA NA
## theta[676] NA NA NA
## theta[677] NA NA NA
## theta[678] NA NA NA
## theta[679] NA NA NA
## theta[680] NA NA NA
## theta[681] NA NA NA
## theta[682] NA NA NA
## theta[683] NA NA NA
## theta[684] NA NA NA
## theta[685] NA NA NA
## theta[686] NA NA NA
## theta[687] NA NA NA
## theta[688] NA NA NA
## theta[689] NA NA NA
## theta[690] NA NA NA
## theta[691] NA NA NA
## theta[692] NA NA NA
## theta[693] NA NA NA
## theta[694] NA NA NA
## theta[695] NA NA NA
## theta[696] NA NA NA
## theta[697] NA NA NA
## theta[698] NA NA NA
## theta[699] NA NA NA
## theta[700] NA NA NA
## theta[701] NA NA NA
## theta[702] NA NA NA
## theta[703] NA NA NA
## theta[704] NA NA NA
## theta[705] NA NA NA
## theta[706] NA NA NA
## theta[707] NA NA NA
## theta[708] NA NA NA
## theta[709] NA NA NA
## theta[710] NA NA NA
## theta[711] NA NA NA
## theta[712] NA NA NA
## theta[713] NA NA NA
## theta[714] NA NA NA
## theta[715] NA NA NA
## theta[716] NA NA NA
## theta[717] NA NA NA
## theta[718] NA NA NA
## theta[719] NA NA NA
## theta[720] NA NA NA
## theta[721] NA NA NA
## theta[722] NA NA NA
## theta[723] NA NA NA
## theta[724] NA NA NA
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# Display posterior information:
plotMCMC(mcmcCoda, data=myData,
zName="Hits", NName="AtBats", sName="Player", cName="PriPos",
compVal=NULL ,
diffCList=list( c("Pitcher", "Catcher"), c("Catcher","1st Base") ),
diffSList=list( c("Kyle Blanks","Bruce Chen"),
c("Mike Leake","Wandy Rodriguez"),
c("Andrew McCutchen", "Brett Jackson"),
c("ShinSoo Choo","Ichiro Suzuki") ),
compValDiff=0.0)
Output 9.11: same as Figure 9.14 left panel
Output 9.12: same as Figure 9.14 right panel
Output 9.13: same as Figure 9.15 left panel
Output 9.14: same as Figure 9.16 left panel
Output 9.15: same as Figure 9.16 right panel
Output 9.16: same as Figure 9.15 right panel
9.5.1 Example: Baseball batting abilities by position book's blog
=> Position-level batting abilities (
=> Abilites of selected individuals with identical batting records
=> Same position with different batting records
Summary of example
shrinkage of individual-ability estimates based on category (fielding position)
: Players with many at-bats (large
than players with few at-bats (small
If not categorize by position,
the estimated abilities of any two players with identical batting records
would also be identical regardless of their positions.
=> The parameter estimates are meaningful descriptions of the data only in the context of the model structure.
9 position parameters (
but dozens or hundreds of players (
=> The certainty of estimate at the overall level is less than the certainty of estimate within each position.
=> If there are only a few categories, the overall level typically is not estimated very precisely.
Test for all the 968 parameters:
In traditional statistical testing based on p-values, we would pay a penalty for even intending to make more comparisons.
In a Bayesian analysis, decisions are based on the posterior distribution, which is based only on the data
(and the prior)
Look for more exercises at https://sites.google.com/site/doingbayesiandataanalysis/