Monte Carlo sampling from the posterior

In Bayesian statistics, we have two sources of information about the parameter 9 our prior belief and the observed data. The prior distribution summarizes our belief about the parameter before we look at the data. The prior density g 9) gives the relative belief weights we have for all possible values of the parameter 0 before we look at the data. In Bayesian statistics, all the information about the parameter 9 that is in the observed data y is contained in the likelihood function f y 9). However, the parameter is considered a random variable, so the likelihood function is written as a conditional distribution. The likelihood function gives the relative support weight each possible value of the parameter 9 has from the observed data. 

Bayes thecM em gives us a unified approach that combines the two sources into a single relative belief weight distribution after we have observed the data. The final belief weight distribution is known as the posterior distribution and it takes into account both the prior belief and the support from the data. Bayes Theorem is usually expressed very simply in the unsealed form posterior propmtional to prior times likelihood. In equation form this is 

This finmula does not give the posterior density g 9 y) exactly, but it does give its shape. In other words, we can find where the modes are, and relative values at any two locations. However, it does not give the scale factor needed to make it a density. This means we cannot calculate probabilities or moments fix)m it. Thus it is not possible to do any inference about the parameter 9 from the unsealed posterior. The 

Understanding Computational Bayesian Statistics. By William M. Bolstad Copyright 2010 John Wiley Sons, Inc. 

The posterior distribution found using Bayes theorem summarizes our knowledge of the parameters given the data that was actually observed. It combines the information from our prior distribution with that from the observed data. A closed form for the integral in the denominator only exists for some particular cases. We will investigate some of these cases in Chapter 4. For other cases the posterior density has to be approximated numerically. This requires integrating 

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The development and implementation of computational methods for drawing random samples from the incompletely known posterior has revolutionized Bayesian statistics. Computational Bayesian statistics breaks free from the limited class of models where the posterior can be found analytically. Statisticians can use observation models, and choose prior distributions that are more realistic, and calculate estimates of the parameters from the Monte Carlo samples from the posterior. Computational Bayesian methods can easily deal with complicated models that have many parameters. This makes the advantages that the Bayesian approach offers accessible to a much wider class of models. 

The computational approach to Bayesian statistics allows the posterior to be approached from a completely different direction. Instead of using the computer to calculate the posterior numerically, we use the computer to draw a Monte Carlo sample from the posterior. Fortunately, all we need to know is the shape of the posterior density, which is given by the prior times the likelihood. We do not need to know the scale factor necessary to make it the exact posterior density. These methods replace the very difficult numerical integration with the much easier process of drawing random samples. A Monte Carlo random sample from the posterior will approximate the true posterior when the sample size is large enough. We will base our inferences on the Monte Carlo random sample from the posterior, not from the numerically calculated posterior. Sometimes this approach to Bayesian inference is the only feasible method, particularly when the parameter space is high dimensional. 

After we have let the chain run a long time, the state the chain is in does not depend on the initial state of the chain. This length of time is called the burn-in period. A draw from the chain after the bum-in time is approximately a random draw from the posterior. However, the sequence of draws from the chain after that time is not a random sample from the posterior, rather it is a dependent sample. In Chapter 3, we saw how we could do inference on the parameters using a random sample from the posterior. In Section 7.3 we will continue with that approach to using the Markov chain Monte Carlo sample from the posterior. We will have to thin the sample so that we can consider it to be approximately a random sample. A chain with good mixing properties will require a shorter burn-in period and less thinning. 

Chapter 6 Markov chain Monte Carlo Sampling from the Posterior... 

A true PPC requires sampling from the posterior distribution of the fixed and random effects in the model, which is typically not known. A complete solution then usually requires Markov Chain Monte Carlo simulation, which is not easy to implement. Luckily for the analyst, Yano, Sheiner, and Beal (2001) showed that complete implementation of the algorithm does not appear to be necessary since fixing the values of the model parameters to their final values obtained using maximum likelihood resulted in PPC distributions that were as good as the full-blown Bayesian PPC distributions. In other words, using a predictive check resulted in distributions that were similar to PPC distributions. Unfortunately they also showed that the PPC is very conservative and not very powerful at detecting model misspecification. 

Parameterization is seen as calibration of the simulator. Sampling from the posterior of parameters is made using Markov Chain Monte Carlo simulation. The sample can be used directly in the Monte Carlo simulation. Here we ran OpenBUGS (Lunn et al., 2009) from R (R Development Core Team, 2008), which makes it possible to parameterize the model and run the simulator in the R environment. 

I. They are not approximations. Estimates found from the Monte Carlo random sample from the posterior can achieve any required accuracy by setting the sample size large enough. 

Markov chain after it has been running a long time it can be considered a random draw from the posterior. This method for drawing a sample from the posterior is known as Markov chain Monte Carlo sampling. 

Sequential draws from a Markov chain are serially dependent. A Markov chain Monte Carlo sample will not be suitable for inference until we have discarded the draws from the burn-in period and thinned the sample so the thinned sample will approximate a random sample from the posterior. In Table A.8 we give the Minitab commands for thinning the output of a MCMC process using the macro ThinMCMC.mac. 

Finally, the Monte Carlo error (MC error) can be used to assess how many iterations need to be run after convergence for accurate inference from the posterior distribution. The MC error is an estimate of the deviance between the mean of the sampled values and the posterior mean this error can be likened to a standard error. Generally, an MC error of less than 5% of the sample standard deviation of the parameters of interest is recommended. 

This is an example of a more general technique called Markov chain Monte-Carlo sampling where, instead of exhaustively searching a state space, one starts from a random state and moves through the space in a stochastic fashion such that, in the limit of long time, each state is visited in proportion to its posterior probability. 

Computational Bayesian statistics is based on drawing a Monte Carlo random sample from the unsealed posterior. This replaces very difficult numerical calculations with the easier process of drawing random variables. Sometimes, particularly for high dimensional cases, this is the only feasible way to find the posterior. 

The great advantage of computational Bayesian methods is that they allow the applied statistician to use more realistic models because he or she is not constrained by analytic or numerical tractability. Models that are based on the underlying situation can be used instead of models based on mathematical convenience. This allows the statistician to focus on the statistical aspects of the model without worrying about calculability. Computational Bayesian methods, based on a Monte Carlo random sample drawn from the posterior, have other advantages as well, even when there are alternatives available. These include ... 

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