Model checking and diagnostics

Convergence Diagnostics

Valid inferences from sequences of MCMC samples are based on the assumption that the samples are derived from the true posterior distribution of interest. Theory guarantees this condition as the number of iterations approaches infinity. It is important, therefore, to determine the minimum number of samples required to ensure a reasonable approximation to the target posterior density. Unfortunately, no universal threshold exists across all problems, so convergence must be assessed independently each time MCMC estimation is performed. The procedures for verifying convergence are collectively known as convergence diagnostics.

One approach to analyzing convergence is analytical, whereby the variance of the sample at different sections of the chain are compared to that of the limiting distribution. These methods use distance metrics to analyze convergence, or place theoretical bounds on the sample variance, and though they are promising, they are generally difficult to use and are not prominent in the MCMC literature. More common is a statistical approach to assessing convergence. With this approach, rather than considering the properties of the theoretical target distribution, only the statistical properties of the observed chain are analyzed. Reliance on the sample alone restricts such convergence criteria to heuristics. As a result, convergence cannot be guaranteed. Although evidence for lack of convergence using statistical convergence diagnostics will correctly imply lack of convergence in the chain, the absence of such evidence will not guarantee convergence in the chain. Nevertheless, negative results for one or more criteria may provide some measure of assurance to users that their sample will provide valid inferences.

For most simple models, convergence will occur quickly, sometimes within a the first several hundred iterations, after which all remaining samples of the chain may be used to calculate posterior quantities. For more complex models, convergence requires a significantly longer burn-in period; sometimes orders of magnitude more samples are needed. Frequently, lack of convergence will be caused by poor mixing. Recall that mixing refers to the degree to which the Markov chain explores the support of the posterior distribution. Poor mixing may stem from inappropriate proposals (if one is using the Metropolis-Hastings sampler) or from attempting to estimate models with highly correlated variables.

Informal Methods

The most straightforward approach for assessing convergence is based on simply plotting and inspecting traces and histograms of the observed MCMC sample. If the trace of values for each of the stochastics exhibits asymptotic behavior over the last $m$ iterations, this may be satisfactory evidence for convergence.

A similar approach involves plotting a histogram for every set of $k$ iterations (perhaps 50-100) beyond some burn in threshold $n$; if the histograms are not visibly different among the sample intervals, this may be considered some evidence for convergence. Note that such diagnostics should be carried out for each stochastic estimated by the MCMC algorithm, because convergent behavior by one variable does not imply evidence for convergence for other variables in the analysis.

An extension of this approach can be taken when multiple parallel chains are run, rather than just a single, long chain. In this case, the final values of $c$ chains run for $n$ iterations are plotted in a histogram; just as above, this is repeated every $k$ iterations thereafter, and the histograms of the endpoints are plotted again and compared to the previous histogram. This is repeated until consecutive histograms are indistinguishable.

Another ad hoc method for detecting lack of convergence is to examine the traces of several MCMC chains initialized with different starting values. Overlaying these traces on the same set of axes should (if convergence has occurred) show each chain tending toward the same equilibrium value, with approximately the same variance. Recall that the tendency for some Markov chains to converge to the true (unknown) value from diverse initial values is called ergodicity. This property is guaranteed by the reversible chains constructed using MCMC, and should be observable using this technique. Again, however, this approach is only a heuristic method, and cannot always detect lack of convergence, even though chains may appear ergodic.

A principal reason that evidence from informal techniques cannot guarantee convergence is a phenomenon called metastability. Chains may appear to have converged to the true equilibrium value, displaying excellent qualities by any of the methods described above. However, after some period of stability around this value, the chain may suddenly move to another region of the parameter space. This period of metastability can sometimes be very long, and therefore escape detection by these convergence diagnostics. Unfortunately, there is no statistical technique available for detecting metastability.

Formal Methods

Along with the ad hoc techniques described above, a number of more formal methods exist which are prevalent in the literature. These are considered more formal because they are based on existing statistical methods, such as time series analysis.

PyMC currently includes three formal convergence diagnostic methods. The first, proposed by Geweke (1992), is a time-series approach that compares the mean and variance of segments from the beginning and end of a single chain.

$$z = \frac{\bar{\theta}_a - \bar{\theta}_b}{\sqrt{S_a(0) + S_b(0)}}$$

where $a$ is the early interval and $b$ the late interval, and $S_i(0)$ is the spectral density estimate at zero frequency for chain segment $i$. If the z-scores (theoretically distributed as standard normal variates) of these two segments are similar, it can provide evidence for convergence. PyMC calculates z-scores of the difference between various initial segments along the chain, and the last 50% of the remaining chain. If the chain has converged, the majority of points should fall within 2 standard deviations of zero.

In PyMC, diagnostic z-scores can be obtained by calling the geweke function. It accepts either (1) a single trace, (2) a Node or Stochastic object, or (4) an entire Model object:

The arguments expected are the following:

• pymc_object: The object that is or contains the output trace(s).

• first (optional): First portion of chain to be used in Geweke diagnostic. Defaults to 0.1 (i.e. first 10% of chain).

• last (optional): Last portion of chain to be used in Geweke diagnostic. Defaults to 0.5 (i.e. last 50% of chain).

• intervals (optional): Number of sub-chains to analyze. Defaults to 20.

The resulting scores are best interpreted graphically, using the geweke_plot function. This displays the scores in series, in relation to the 2 standard deviation boundaries around zero. Hence, it is easy to see departures from the standard normal assumption.

The second diagnostic provided by PyMC is the Raftery and Lewis (1992) procedure. This approach estimates the number of iterations required to reach convergence, along with the number of burn-in samples to be discarded and the appropriate thinning interval. A separate estimate of both quantities can be obtained for each variable in a given model.

As the criterion for determining convergence, the Raftery and Lewis approach uses the accuracy of estimation of a user-specified quantile. For example, we may want to estimate the quantile $q=0.975$ to within $r=0.005$ with probability $s=0.95$. In other words,

$$Pr(|\hat{q}-q| \le r) = s$$

From any sample of $\theta$, one can construct a binary chain:

$$Z^{(j)} = I(\theta^{(j)} \le u_q)$$

where $u_q$ is the quantile value and $I$ is the indicator function. While $\{\theta^{(j)}\}$ is a Markov chain, $\{Z^{(j)}\}$ is not necessarily so. In any case, the serial dependency among $Z^{(j)}$ decreases as the thinning interval $k$ increases. A value of $k$ is chosen to be the smallest value such that the first order Markov chain is preferable to the second order Markov chain.

This thinned sample is used to determine number of burn-in samples. This is done by comparing the remaining samples from burn-in intervals of increasing length to the limiting distribution of the chain. An appropriate value is one for which the truncated sample's distribution is within $\epsilon$ (arbitrarily small) of the limiting distribution. Estimates for sample size tend to be conservative.

This diagnostic is best used on a short pilot run of a particular model, and the results used to parameterize a subsequent sample that is to be used for inference.

The arguments are:

• pymc_object: The object that contains the Geweke scores. Can be a list (one set) or a dictionary (multiple sets).

• q: Desired quantile to be estimated.

• r: Desired accuracy for quantile.

• s (optional): Probability of attaining the requested accuracy (defaults to 0.95).

• epsilon (optional) : Half width of the tolerance interval required for the q-quantile (defaults to 0.001).

The third convergence diagnostic provided by PyMC is the Gelman-Rubin statistic Gelman and Rubin (1992). This diagnostic uses multiple chains to check for lack of convergence, and is based on the notion that if multiple chains have converged, by definition they should appear very similar to one another; if not, one or more of the chains has failed to converge.

The Gelman-Rubin diagnostic uses an analysis of variance approach to assessing convergence. That is, it calculates both the between-chain varaince (B) and within-chain varaince (W), and assesses whether they are different enough to worry about convergence. Assuming $m$ chains, each of length $n$, quantities are calculated by:

$$\begin{align}B &= \frac{n}{m-1} \sum_{j=1}^m (\bar{\theta}_{.j} - \bar{\theta}_{..})^2 \\ W &= \frac{1}{m} \sum_{j=1}^m \left[ \frac{1}{n-1} \sum_{i=1}^n (\theta_{ij} - \bar{\theta}_{.j})^2 \right] \end{align}$$

for each scalar estimand $\theta$. Using these values, an estimate of the marginal posterior variance of $\theta$ can be calculated:

$$\hat{\text{Var}}(\theta | y) = \frac{n-1}{n} W + \frac{1}{n} B$$

Assuming $\theta$ was initialized to arbitrary starting points in each chain, this quantity will overestimate the true marginal posterior variance. At the same time, $W$ will tend to underestimate the within-chain variance early in the sampling run. However, in the limit as $n \rightarrow \infty$, both quantities will converge to the true variance of $\theta$. In light of this, the Gelman-Rubin statistic monitors convergence using the ratio:

$$\hat{R} = \sqrt{\frac{\hat{\text{Var}}(\theta | y)}{W}}$$

This is called the potential scale reduction, since it is an estimate of the potential reduction in the scale of $\theta$ as the number of simulations tends to infinity. In practice, we look for values of $\hat{R}$ close to one (say, less than 1.1) to be confident that a particular estimand has converged. In PyMC, the function gelman_rubin will calculate $\hat{R}$ for each stochastic node in the passed model:

For the best results, each chain should be initialized to highly dispersed starting values for each stochastic node.

By default, when calling the summary_plot function using nodes with multiple chains, the $\hat{R}$ values will be plotted alongside the posterior intervals.

Goodness of Fit

Checking for model convergence is only the first step in the evaluation of MCMC model outputs. It is possible for an entirely unsuitable model to converge, so additional steps are needed to ensure that the estimated model adequately fits the data. One intuitive way of evaluating model fit is to compare model predictions with the observations used to fit the model. In other words, the fitted model can be used to simulate data, and the distribution of the simulated data should resemble the distribution of the actual data.

Fortunately, simulating data from the model is a natural component of the Bayesian modelling framework. Recall, from the discussion on imputation of missing data, the posterior predictive distribution:

$$p(\tilde{y}|y) = \int p(\tilde{y}|\theta) f(\theta|y) d\theta$$

Here, $\tilde{y}$ represents some hypothetical new data that would be expected, taking into account the posterior uncertainty in the model parameters. Sampling from the posterior predictive distribution is easy in PyMC. The code looks identical to the corresponding data stochastic, with two modifications: (1) the node should be specified as deterministic and (2) the statistical likelihoods should be replaced by random number generators. Consider the gelman_bioassay example, where deaths are modeled as a binomial random variable for which the probability of death is a logit-linear function of the dose of a particular drug.

The posterior predictive distribution of deaths uses the same functional form as the data likelihood, in this case a binomial stochastic. Here is the corresponding sample from the posterior predictive distribution:

Notice that the observed stochastic Binomial has been replaced with a stochastic node that is identical in every respect to `deaths`, except that its values are not fixed to be the observed data -- they are left to vary according to the values of the fitted parameters.

The degree to which simulated data correspond to observations can be evaluated in at least two ways. First, these quantities can simply be compared visually. This allows for a qualitative comparison of model-based replicates and observations. If there is poor fit, the true value of the data may appear in the tails of the histogram of replicated data, while a good fit will tend to show the true data in high-probability regions of the posterior predictive distribution. The Matplot package in PyMC provides an easy way of producing such plots, via the gof_plot function.

A second approach for evaluating goodness of fit using samples from the posterior predictive distribution involves the use of a statistical criterion. For example, the Bayesian p-value (Gelman et al. 1996) uses a discrepancy measure that quantifies the difference between data (observed or simulated) and the expected value, conditional on some model. One such discrepancy measure is the Freeman-Tukey statistic (Brooks et al. 2000):

$$D(x|\theta) = \sum_j (\sqrt{x_j}-\sqrt{e_j})^2,$$

where the $x_j$ are data and $e_j$ are the corresponding expected values, based on the model. Model fit is assessed by comparing the discrepancies from observed data to those from simulated data. On average, we expect the difference between them to be zero; hence, the Bayesian p value is simply the proportion of simulated discrepancies that are larger than their corresponding observed discrepancies:

$$p = Pr[ D(x_{\text{sim}}|\theta) > D(x_{\text{obs}}|\theta) ]$$

If $p$ is very large (e.g. $>0.975$) or very small (e.g. $\lt 0.025$) this implies that the model is not consistent with the data, and thus is evidence of lack of fit. Graphically, data and simulated discrepancies plotted together should be clustered along a 45 degree line passing through the origin.

The discrepancy function in the diagnostics package can be used to generate discrepancy statistics from arrays of data, simulated values, and expected values:

D = pymc.discrepancy(x, x_sim, x_exp)

For a dataset of size $n$ and an MCMC chain of length $r$, this implies that x is size (n,), x_sim is size (r,n) and x_exp is either size (r,) or (r,n). A call to this function returns two arrays of discrepancy values (simulated and observed), which can be passed to the discrepancy_plot function in the `Matplot` module to generate a scatter plot, and if desired, a p value:

Exercise: Meta-analysis of beta blocker effectiveness

Carlin (1992) considers a Bayesian approach to meta-analysis, and includes the following examples of 22 trials of beta-blockers to prevent mortality after myocardial infarction.

In a random effects meta-analysis we assume the true effect (on a log-odds scale) $d_i$ in a trial $i$ is drawn from some population distribution. Let $r^C_i$ denote number of events in the control group in trial $i$, and $r^T_i$ denote events under active treatment in trial $i$. Our model is:

$$\begin{aligned} r^C_i &\sim \text{Binomial}\left(p^C_i, n^C_i\right) \\ r^T_i &\sim \text{Binomial}\left(p^T_i, n^T_i\right) \\ \text{logit}\left(p^C_i\right) &= \mu_i \\ \text{logit}\left(p^T_i\right) &= \mu_i + \delta_i \\ \delta_i &\sim \text{Normal}(d, t) \\ \mu_i &\sim \text{Normal}(m, s) \end{aligned}$$

We want to make inferences about the population effect $d$, and the predictive distribution for the effect $\delta_{\text{new}}$ in a new trial. Build a model to estimate these quantities in PyMC, and (1) use convergence diagnostics to check for convergence and (2) use posterior predictive checks to assess goodness-of-fit.

Here are the data:

References

Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science. A Review Journal of the Institute of Mathematical Statistics, 457–472.

Geweke, J., Berger, J. O., & Dawid, A. P. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics 4.

Brooks, S. P., Catchpole, E. A., & Morgan, B. J. T. (2000). Bayesian Animal Survival Estimation. Statistical Science. A Review Journal of the Institute of Mathematical Statistics, 15(4), 357–376. doi:10.1214/ss/1177010123

Gelman, A., Meng, X., & Stern, H. (1996). Posterior predicitive assessment of model fitness via realized discrepencies with discussion. Statistica Sinica, 6, 733–807.

Raftery, A., & Lewis, S. (1992). One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo. Statistical Science. A Review Journal of the Institute of Mathematical Statistics, 7, 493–497.

CrossValidated: How to use scikit-learn's cross validation functions on multi-label classifiers