Markov Chain Monte Carlo Sampling from Posterior

Markov Chain Monte Carlo Sampling from Posterior 

Surprisingly MCMC methods are more efficient than the direct procedures for drawing samples from the posterior when we have a large number of parameters. We run the chain long enough so that it approaches the long-run distribution. A value taken after that is a draw from the long-run distribution. 

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

We want to find a Markov chain that has the posterior distribution of the parameters given the data as its long-run distribution. Thus the parameter space will be the state space of the Markov chain. We investigate how to find a Markov chain that satisfies this requirement. We know that the long-run distribution of a ergodic Markov chain is a solution of the steady state equation. That means that the long-run distribution 7T of a finite ergodic Markov chain with one-step transition matrix P satisfies the equation 

This says the long-run probability of a state equals the weighted sum of one-step probabilities of entering that state from all states each weighted by its long-run probability. The comparable steady state equation that ir 0), the long-run distribution of a Markov chain with a continuous state space, satisfies is given by 

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MARKOV CHAIN MONTE CARLO SAMPLING FROM POSTERIOR... 

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... 

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. 

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. 

Markov chain Monte Carlo samples are not independent random samples. This is unlike the case for samples drawn directly from the posterior by acceptance-rejection sampling. This means that it is more difficult to do inferences from the Markov chain Monte carlo sample. In this section we discus the differing points of view on this problem. 

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. 

Population models describe the relationship between individuals and a population. Individual parameter sets are considered to arise from a joint population distribution described by a set of means and variances. The conditional dependencies among individual data sets, individual variables, and population variables can be represented by a graphical model, which can then be translated into the probability distributions in Bayes theorem. For most cases of practical interest, the posterior distribution is obtained via numerical simulation. It is also the case that the complexity of the posterior distribution for most PBPK models is such that standard MC sampling is inadequate leading instead to the use of Markov Chain Monte Carlo (MCMC) methods... 

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. 

Variability measures The likelihood functions and prior distributions have been incorporated in a Bayesian inference procedure in which the posterior density n 01E) is computed. The Bayesian inference is performed by using a Markov Chain Monte Carlo Method (MCMC), which allows samples to be generated from a continuous unnormaUzed density (Chiband Greenberg, 1995). The MCMC method, which is frequently applied to Bayesian Inference problems (Gilks et al. 1996), results in a m-samples set S = 0,. .., 0, representing the... 

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. 

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