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Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review By Mary Kathryn Cowles and Bradley P. Carlin Presented by Yuting Qi 12/01/2006

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OUTLINE MCMC Convergence Diagnostics: Introduce 4 Methods in details Focus on Prescriptive summary Underlying theoretical basis Advantages and disadvantages Comparative results

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1. Gelman and Rubin (1992) 1/4 What ? Based on normal theory approximation to exact Bayesian posterior inference: Focus on applied inference for Bayesian posterior distributions in real problem, which often tend toward normality after transformations and marginalization. Two major steps: Create an overdispersed estimate of the target distribution and use it to start several independent sequences. Analyze the multiple sequences to form a distributional estimate of what is known about the target r.v. given the simulations so far. The distributional estimate is a Student’s t distribution of each scalar quantity of interest. Convergence: Convergence is monitored by estimating the factor by which the scale parameter might shrink for infinite sampling.

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1. Gelman and Rubin (1992) 2/4 How ? Step 1: Creating a starting distribution Locate the high-density regions of the target distribution of x and find the K modes. Approximate the high-density regions by a GMM: Form an overdispersed distribution by first drawing from the GMM and then dividing each sample by a positive number, which results in a mixture t distributions: Sharpen the overdispersed approximation by downweighting regions that have relatively low density through importance resampling for example.

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1. Gelman and Rubin (1992) 3/4 Step 2: Re-estimating the target distributions Independently simulate m sequences of length 2 n from the overdispersed distribution and discard the first n iterations. For each scalar parameter of interest, estimate the following quantity from the last n iterations of m sequences: B : the variance between the means from m sequences; W : the average of the m within-sequence variances; : estimate of target mean: mean of mn samples : estimate of target variance (unbiased): Estimate the posterior of target distribution as a t distribution (considering variability of the estimates and ) with center and scale. Monitor the convergence by shrink factor, as it near 1 for all scalars, collect burn-out samples.

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1. Gelman and Rubin (1992) 4/4 Comments: approaches to 1: within-sequences variance dominant between-sequences variance, all sequences escaped the influence of starting points and traverse all target distributions. Quantitative. Criticisms: Rely on the user’s ability to find a start distribution. Rely on normal approximation for diagnosing convergence to the true posterior. Inefficient, multiple sequences and discard a large number of early iterations.

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2. Geweke (1992) 1/3 What ? Use methods from spectral analysis to assess convergence and the intent is to estimate the mean E[g( )] of some function g( ) of interest. Collect g( (j) ) after each iteration Treat { g( (j) ) } j=1,p as time series and compute spectral density S G ( ). Use numerical standard error (NSE) and relative numerical efficiency (RNE) to monitor convergence. Assumption: The MCMC process and the importance function g( ), jointly imply the existence of a spectrum, and the existence of a spectral density with no discontinuities at the frequency 0.

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2. Geweke (1992) 2/3 How ? Estimate E[g( )] from p iterations Asymptotically estimator: Asymptotic variance: Determine preliminary iterations: Given the sequence {G(j)} j=1,p, if {G(j)} is stationary, as p->inf Determine sufficient iterations: Numerical standard error (NSE): Relative numerical efficiency (RNE) Indicating the number of draws wound be required to produce the same numerical accuracy if the draws had been made from an iid sample drawn directly from the posterior distribution. 0 1

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2. Geweke (1992) 3/3 Comments Address the issues of both bias and variance. Is univariate. Require a single sampler chain. Disadvantages: Is sensitive to the spectral window. Not specify a procedure for applying the diagnostic but leave to the subjective choice of the users.

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3. Ritter and Tanner (1992) 1/3 The Gibbs Stopper Convert the output of the Gibbs sampler to a sample from the exact distribution. Assign a weight w to the d-dimensional vector X drawn from the current iteration: q is a function proportional to the joint distribution; g i is the current Gibbs sampler approximation. Assess the convergence: If the current approximation to the joint distribution is “close” to the true one, then the distribution of the weights will be degenerate about a constant.

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3. Ritter and Tanner (1992) 2/3 Compute g i : Let The joint distribution of the samples obtained at iteration i +1 is g i+1 ( X ) = The integration can be approximated by Monte Carlo method g i+1 ( X ) X 1, …, X m are samples drawn at iteration i. Probability of moving from X’ (at iteration i) to X at iteration i+1.

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3. Ritter and Tanner (1992) 3/3 Comments: Assess distributional convergence; Disadvantages: Applicable only with the Gibbs sampler; Coding is problem-specific; Computation of weights can be time-intensive; If full conditionals are not standard distributions, we must estimate the normalizing constants.

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4. Zellner and Min (1995) 1/3 Gibbs Sampler Convergence Criteria (GSC 2 ) Aim to determine whether the Gibbs sampler not only has converged, but also has converged to a correct result. Divide the model parameters into two parts , Derive analytical forms for Three convergence criterions: Assume ( 1, 1 ) and ( 2, 2 ) are two points in the parameter space Easily sampled posterior conditionals priorlikelihood

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4. Zellner and Min (1995) 2/3 1. The anchored ratio convergence criterion (ARC 2 ) Calculate: If the Gibbs sampler output is “satisfactory”, then and will be close to. 2. The difference convergence criterion (DC 2 ) Since If ->0, then “satisfactory” 3. The ratio convergence criterion (RC 2 ) If ->1, then “satisfactory” Ratio of joint posterior density for the selected two points

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4. Zellner and Min (1995) 3/3 Comments: Quantitative; Require a single sampler chain; Coding is problem-specific and analytical work is needed; Disadvantage: Application is limited when the factorization cannot be achieved.

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Comparative results 1/3 Trivariate Normal with high correlations Run the samplers for relatively few iterations to test these methods detect convergence failure or ambiguity.

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Comparative results 2/3 1. Gelman & Rubin: shrink factors (->1) 2. Geweke: NSE (->0)

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Comparative results 3/4 Ritter & Tanner: Gibbs stopper (weights w -> constant)

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Comparative results 4/4 Zellner & Min: Difference convergence Criterion ( -> 0)

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Comparative results 5/5 Remarks: Geweke’s diagnostic appears to be premature; Gelman & Rubin’s method may be consistent with the fact; however choosing the starting points is critical; The results of other methods are difficult to interpret.

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Summary, Discussion, and Recommendation Be cautious when using these diagnostics; Use a variety of diagnostic tools rather than any single one; Learn as much as possible about the target density before applying MCMC algorithm;

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