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1 parent e203db4 commit dafe9ebf337fca4746debad0425df6d182953799 @cboettig committed Mar 27, 2013
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  1. +134 −56 inst/doc/nonparametric-bayes.Rmd
@@ -51,59 +51,88 @@ simple mechanistic models can nevertheless provide imporant insights
into possible dynamics -- for instance, demonstrating that a vaccine
does not have to be 100% effective to eliminate the transmission of a
virus [@Kermack1921] -- such approaches are not well-suited for use in
-forecasting future dynamics upon which outcomes management policy can
-be based. Despite a long history of work emphasizing the difference
-between modeling for understanding (generic or strategic models)
-and modeling for prediction (precise, specific, or tactical models)
-[e.g. @Levins1966], simple parametric models continue to find application
-in predictive management contexts. The management of marine fisheries
-is a prime example, in which simple, mechanistically motivated models of
-stock-recruitment relationships such as Ricker or Beverton-Holt curves
-are systematically fit to data and used in solving decision-theoretic
-or optimal control problems determining resource management policies
-such as maximum sustainable yield [@refs].
-
-<!-- Better defintion of focal problems here .... -->
-Stochastic dynamic programming @Mangel1982
-
-To address structural uncertainty in the processes involved, such mechanistic models
-can be adapted to accomidate uncertainty in the parameters or consider a set of
-alternative models simultaneously (belief SDP)
-
-
+forecasting outcomes of potential management options. Non-parametric
+approaches offer a more flexible alternative that can both more accurately
+reflect the data available while also representing greater uncertainty
+in areas (of state-space) where data is lacking. Ecological research
+and management strategy should pay closer attention to the opportunities
+and challenges nonparametric modeling can offer.
+
+
+### Quantitative vs Qualitative Decisions
+
+In this paper, we consider those ecological management problems in which
+a mathematical (or computational) model is used to quantitatively inform
+decision-making by allowing a manager to compare to the expected consequences
+of potential management actions (or policies). We distinguish this from the
+solely qualitative use of a model, in which models are used to represent and
+compare hypotheses of different mechanisms that would lead to qualitatively
+different actions.
+
+We argue that while simple mechanistically motivated models
+may be best for the latter case [@Cunningham2013; @Geritz2012], such models
+can be not only inaccurate but misleading in quantitative decision making.
+Nonparametric models can more realistically represent uncertainties outside
+of observed data while also better capturing the dynamics in the region
+observed. Advances in the theory and computational implementations of
+nonparametric methods make them ripe for such applications. We use the classic
+problem of optimal harvest of a marine fishery to illustrate how the
+nonparametric approach of Gaussian processes can be applied.
+
+We imagine the manager seeks to maximize some known socio-economic value function,
+given their knowledge of the system.
+In particular, we will focus on the case in which the manager faces a series of decisions,
+such as setting a harvest quota for a marine fishery each year based on observations
+of the stock size the year before.
+
+
+### Uncertainties: the classical approach
+
+The ecological dynamics of most systems of management interest are
+typically both complex and unknown [@refs]. Despite this, quantitative policy
+optimization in these systems has almost exclusively been based on
+simple, mechanistically motivated parametric models.
+There are several reasons for this. First, limited available data almost
+always percludes any accurate estimate of more complicated models. Second,
+the computational demands of determining an optimal stategy in a sequential
+decision-theory problem suffer from the "curse of dimensionality" that makes
+it difficult to consider more complicated models, states, or actions. Unfortunately,
+these challenges also make it particularly difficult to reflect the true uncertainty
+either in the parameters of the given models (parametric uncertainty) or in
+the structure of the models themselves (structural uncertainty).
Further unknowns such as measurement uncertainty, parameter uncertainty,
unobserved states, knowledge of boundary conditions, etc. further
compound the issue. Though a hierarchical Bayesian approach provides a
natural way to address these from a statistical standpoint, formulating
-reasonable parametric descriptions of each form of uncertainty is a
+reasonable parametric descriptions of each form of uncertainty is a
challenging task in itself, let alone the computational difficulties
of solving such a system. @Cressie2009 provides a good account of the
successess and challenges of the approach. Applying these approaches in
the management context of sequential decision making, in which forecasts
must be obtained over a range of possible actions and updated regularly
as new information arrives makes such an approach less feasible still.
+
+An active literature and growing computational power over the past several decades have
+only marginally improved this situation. Parametric or structural uncertainty
+can be introduced only by increasing the state space to reflect a distribution
+of possible parameters [@refs] or the degree of belief in each of a set of possible models [@refs]
+Frequently, the space of possible actions must then be reduced or the algorithms adjusted
+by approximations to keep computations feasible.
+
+
Though machine learning approaches have begun to appear in the
ecological and conservation literature (Species distribution models),
including the Gaussian process based approach used here [@Munch2005],
they remain unfamiliar and untrusted approaches for most ecologists.
-Machine learning approaches represent an essentially pattern-based
-rather than process-based approach, raising the same skepticism from most
-theoretical ecologists that they hold for older correlative methods such
-as linear regression, while the complexity of the techniques has barred
-their adoption in more empirical audiences. <!-- Whoa! rampant speculation
-alert! -->
-
-<!-- Whoa, isn't machine learning used to make decisions all the time,
- eg, in engineering and Artificial Intelligence? -->
-
-A more immediate barrier to their adoption is the absence of a framework
+One potential barrier to their adoption is the absence of a framework
for applying machine learning approaches to resource management problems.
Traditional approaches to optimal control (Pontryagin's principle, stochastic
dynamic programming) rely on knowledge of the state equation, usually described
by a simple parametric model. Here we illustrate how a stochastic dynamic
programming algorithm can alternatively be driven by the predictions from
a Gaussian process -- a machine learning approximation to the state dynamics.
+
<!-- Okay, perhaps that's novel, but it's pretty trivial. Isn't it
obvious to everyon that it's trivial? -->
@@ -125,34 +154,49 @@ parameter space that have been poorly or never sampled.
Approach and Methods
=====================================================================
+### The fisheries management problem
-```{r stateeq}
-f <- RickerAllee
-p <- c(2, 10, 5)
-K <- 10
-allee <- 5
-```
+In our example, we focus on the problem in which a manager must set
+the harvest level for a marine fishery each year to maximize the net
+present value of the resource, given an estimated stock size from the year
+before. Rich data and global concerns have made marine fisheries the crucible for much
+of the founding work [@Gordon1954; @Reed1979; @May1979; @Ludwig1982]
+in managing ecosystems under uncertainty. Global trends [@Worm2006]
+and controversy [@Hilborn2007; @Worm2009] have made understanding these
+challenges all the more pressing.
-```{r sdp-pars, dependson="stateeq"}
-sigma_g <- 0.05
-sigma_m <- 0.0
-z_g <- function() rlnorm(1, 0, sigma_g)
-z_m <- function() 1+(2*runif(1, 0, 1)-1) * sigma_m
-x_grid <- seq(0, 1.5 * K, length=101)
-h_grid <- x_grid
-profit <- function(x,h) pmin(x, h)
-delta <- 0.01
-OptTime <- 20 # stationarity with unstable models is tricky thing
-reward <- 0
-xT <- 0
-seed_i <- 1
-Xo <- K # observations start from
-x0 <- Xo # simulation under policy starts from
-Tobs <- 35
-```
+To permit comparisons against a theoretical optimum we will consider
+data on the stock dynamics simulated from a simple parametric model
+in which recruitment of the fish stock $X_{t+1}$ in the following year
+is a stochastic process governed by a function $f$ of the current
+stock $X_t$, selected harvest policy $h_t$, and noise process $Z$,
+
+$$X_{t+1} = Z_t f(X_t, h_t) $$
+
+
+Given parameters for the function $f$ and probability distribution $Z$,
+along with a given economic model determining the price/profit $\Pi(X_t, h_t)$
+realized in a given year given a choice of harvest $h_t$ and observed stock $X_t$.
+This problem can be solved exactly for discretized values of stock $X$ and policy $h$
+using stochastic dynamic programming [@Mangel1982].
+Problems of this sort underpin much marine fisheries management today.
+
+A crux of this approach is correctly specifying the functional form of $f$,
+along with its parameters. The standard approach [@refs] uses one of a
+handful of common parametric models representing the stock-recruitment
+relationship, usually after estimating the model parameters from any
+available existing data. Though modifications of this approach mentioned in the
+introduction can permit additional sources of uncertainty such as measurement
+error in the stock assessment, implementation errors in the harvest policy, [@Clark1986, @Roughgarden1996, @Sethi2005]
+uncertainty in parameters [@Mangel1985, @Schaupaugh2013] or model structure [@Williams2001, @Athanassoglou2012]
-### Background on Gaussian Process inference
+
+Here, we compare this approach to our alternative
+that uses a Gaussian Process in place of the a given stock recruitment curve.
+
+### The Non-parametric Bayesian alternative for stock-recruitment curves
+
The use of Gaussian process regression (known as Kreging in the geospatial
literature) to formulate a predictive model is relatively new in the
@@ -182,12 +226,46 @@ curves that can be though of as approximations to a range of possible
to consider a set of possible curves simultaneously.
<!-- Figure 1 include curves drawn from the posterior density? -->
+Following [@Munch2005], we will estimate Gaussian processes from
+
<!-- Do we need more specifics on Gaussian process as an approximation
to parametric models? Discussion of Gaussian process vs other machine
learning / forecasting approaches that have less clear statistical
foundations? If so, does this all belong in the discussion? -->
+###
+
+
+```{r stateeq}
+f <- RickerAllee
+p <- c(2, 10, 5)
+K <- 10
+allee <- 5
+```
+
+```{r sdp-pars, dependson="stateeq"}
+sigma_g <- 0.05
+sigma_m <- 0.0
+z_g <- function() rlnorm(1, 0, sigma_g)
+z_m <- function() 1+(2*runif(1, 0, 1)-1) * sigma_m
+x_grid <- seq(0, 1.5 * K, length=101)
+h_grid <- x_grid
+profit <- function(x,h) pmin(x, h)
+delta <- 0.01
+OptTime <- 20 # stationarity with unstable models is tricky thing
+reward <- 0
+xT <- 0
+seed_i <- 1
+Xo <- K # observations start from
+x0 <- Xo # simulation under policy starts from
+Tobs <- 35
+```
+
+
+### Background on Gaussian Process inference
+
+
### The optimal control problem

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