Diagnosing MCMC Near Decision Boundaries

Why Decisions Near a Cutoff Need Special Diagnostics

A posterior decision rule often compares a posterior probability with a prespecified cutoff. For example, an arm might be retained if the posterior probability that its treatment effect exceeds a clinically relevant margin is greater than 0.90. If the estimated probability is 0.98, modest simulation instability is unlikely to change that rule. If it is 0.901, a small computational shift could reverse the decision. The nearer the estimate is to the cutoff, the more important it is to establish that the MCMC output has adequately explored the decision-relevant part of the posterior.

The issue is not simply whether the parameter chains look smooth or whether a long run produced many draws. A decision is based on a particular posterior event, such as an arm exceeding a margin or having the highest utility. The chain must estimate the probability of that event reliably. Diagnostics should therefore focus on the event that defines the decision, as well as on relevant model quantities.

This discussion assumes that the posterior model and the event \(D\) have been specified correctly, that the MCMC transition method targets that posterior, and that retained draws are intended to represent its stationary distribution. Convergence diagnostics address computational sampling; they do not establish that the likelihood, prior, or decision rule is scientifically appropriate. The earlier material on computing decision probabilities with MCMC gives the basic estimate: evaluate the event at each retained draw and average the resulting indicators.

Diagnose the Quantity That Drives the Decision

For retained draws \(\theta^{(1)},\ldots,\theta^{(M)}\), define \(I_m=1\) if draw \(m\) satisfies \(D\), and \(I_m=0\) otherwise. The MCMC estimate is \(\hat q=M^{-1}\sum_{m=1}^M I_m\). The sequence \(I_1,\ldots,I_M\) is itself a chain-derived sequence. Its behavior can differ sharply from that of a treatment-effect parameter: a parameter may move smoothly while the decision event changes rarely, or a parameter chain may appear stable while the event indicator is nearly constant within individual chains.

Use several chains initialized from suitably dispersed starting points, discard warmup according to the sampling procedure, and inspect retained draws. For a boundary-sensitive decision, inspect the event indicator and the quantities defining the event: for instance, a treatment-control contrast and its margin. A useful diagnostic set includes trace plots, chain-specific event proportions, autocorrelation or mixing patterns, effective sample size (ESS), and a between-chain convergence diagnostic.

• Trace plots: Look for chains that explore comparable ranges without persistent drift, long stuck segments, or distinct chain-specific regions. Plot the decision indicator or event-defining contrast, not just convenient model parameters.
• Between-chain agreement: Compare each chain’s estimate of \(q\). Large differences can reveal that pooled draws conceal poor exploration.
• Rank-normalized split R-hat: This compares within-chain and between-chain variation after splitting chains and applying rank-based normalization. Values near 1 indicate better agreement; values above about 1.01 are a warning in many workflows, not a universal pass/fail rule.
• Event-specific ESS: Calculate or inspect ESS for \(I_m\), as well as for relevant contrasts. A large ESS for a treatment-effect parameter does not guarantee a large ESS for an indicator of a threshold event.

A basic ESS approximation for a scalar chain of length \(M\) is \(M/[1+2\sum_{k=1}^{\infty}\rho_k]\), where \(\rho_k\) is its lag-\(k\) autocorrelation. This motivates why correlated draws provide less information than the same number of independent draws. For a cutoff decision, apply the idea to the indicator sequence: if exceedances cluster into long runs, the indicator is autocorrelated and its effective sample size can be much smaller than the nominal number of draws.

Why Event-Specific ESS Matters

The indicator definition gives a direct justification. Each draw contributes either zero or one to \(\hat q\), so the decision-probability estimate depends on the frequency and clustering of those two values. Autocorrelation in \(I_m\) determines how much the sequence’s repeated exceedances or non-exceedances reduce the information in the run. A parameter’s ESS instead describes the sampling behavior of that parameter; it is not automatically the ESS of a nonlinear event defined by several parameters.

This expression is a diagnostic summary, not a guarantee that a decision is safe. In particular, if an event is very rare, most indicators may be zero. A chain that never visits the event can give a deceptively simple-looking indicator trace, and a conventional ESS calculation may be uninformative or undefined when the indicator has no variation. Such output does not establish that the posterior probability is zero; it may indicate that the sampler has not adequately visited the relevant region.

Use Diagnostics Together, Not as a Single Gate

No one diagnostic certifies convergence. R-hat can be close to one when all chains are trapped in the same region. Conversely, a high R-hat is a warning that chains do not yet agree, not a direct measure of how often a decision would change. Trace plots can expose sticking or drifting, but visual inspection alone is subjective. ESS reflects autocorrelation under the sampled chain, but cannot reveal an important region that no chain has visited.

Interpret diagnostics in light of the decision margin \(|\hat q-c|\). When that margin is small, modest chain disagreement or a low event-specific ESS deserves particular attention. If the estimate is far from the cutoff and chains mix well for the event, the computational decision may be more robust. However, “far” is relative: it should be assessed against the variability and stability of the event estimates, not by a fixed universal distance.

For multiple candidate arms, identify every probability or comparison that could change the action. An arm may be retained by exceeding an efficacy probability cutoff, dropped by falling below a futility cutoff, or selected by having the largest utility. Diagnose the corresponding indicator or ranking event for each decision-relevant quantity. Checking only one arm’s effect chain can miss poor exploration of a joint event such as “arm A has the highest utility.”

Practical Responses and Alternatives

Longer runs can help when chains are mixing adequately but estimates remain unstable. They are not a remedy for chains trapped in separated regions. Reparameterization or a more suitable transition method may improve exploration. For some models, an analytic calculation or numerical integration can provide a useful independent check; whether such an approach is available depends on the posterior and event. These alternatives should be assessed for their own assumptions rather than treated as automatic replacements for diagnostics.

A common mistake is to pool a large number of draws, see \(\hat q\) barely exceed \(c\), and declare the arm eligible. That reasoning treats correlated draws as independent and ignores whether chains agree on the event. The correction is to inspect the decision indicator across chains, assess its mixing and ESS, and treat a near-boundary result with unresolved diagnostics as computationally inconclusive. This is a statement about the adequacy of the calculation, not a reason to alter the prespecified cutoff after seeing the output.