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Examples; Eyes: Normal Mixture Model

Eyes: Normal Mixture Model

An example from OpenBUGS openbugs:2014:ex, Bowmaker bowmaker:1985:TTT, and Robert robert:1994:MDI concerning 48 peak sensitivity wavelength measurements taken on a set of monkey's eyes.

Model

Measurements are modelled as the mixture distribution

$$\begin{aligned} y_i &\sim \text{Normal}(\lambda_{T_i}, \sigma) \quad\quad i=1,\ldots,48 \\\ T_i &\sim \text{Categorical}(P) \\\ \lambda_1 &\sim \text{Normal}(0, 1000) \\\ \lambda_2 &= \lambda_1 + \theta \\\ \theta &\sim \text{Uniform}(0, 1000) \\\ \sigma^2 &\sim \text{InverseGamma}(0.001, 0.001) \\\ P &= \text{Dirichlet}(1, 1) \end{aligned}$$

where yi is the measurement on monkey i.

Analysis Program

eyes.jl

Results

Iterations = 2502:10000
Thinning interval = 2
Chains = 1,2
Samples per chain = 3750

Empirical Posterior Estimates:
              Mean         SD       Naive SE       MCSE         ESS
     P[1]   0.60357102 0.08379221 0.0009675491 0.0013077559 3750.00000
     P[2]   0.39642898 0.08379221 0.0009675491 0.0013077559 3750.00000
       s2  14.45234459 4.96689604 0.0573527753 0.1853970211  717.73584
lambda[1] 536.75250337 0.88484533 0.0102173137 0.0304232594  845.90821
lambda[2] 548.98693469 1.18938418 0.0137338256 0.0625489045  361.58042

Quantiles:
              2.5%         25.0%        50.0%        75.0%        97.5%
     P[1]   0.43576584   0.54842468   0.60395494   0.66066383   0.76552882
     P[2]   0.23447118   0.33933617   0.39604506   0.45157532   0.56423416
       s2   8.55745263  11.37700782  13.39622105  16.26405571  27.08863166
lambda[1] 535.08951002 536.16891895 536.73114152 537.30710397 538.61179506
lambda[2] 546.57859195 548.24188698 548.97732892 549.74377421 551.38780148