Why Posterior Shape Matters
A Bayesian joint model can produce a posterior distribution that is difficult to summarize with one estimate and one interval. The posterior may have separate regions of high probability, or it may be broad along a parameter direction because the observed data provide little information there. These are different features: multimodality concerns multiple distinct high-density regions, while weak identification concerns limited information about one or more parameters. Either can make posterior summaries sensitive to how the model is explored and to the prior.
This issue is especially relevant for shared random-effects models. The longitudinal measurements and event-time outcome jointly inform latent subject effects and association parameters, but the data may not distinguish all parameter combinations equally well. The earlier material on likelihood ridges and weak association identification provides related likelihood-based context. Here the focus is on what the posterior and its sampling chains reveal.
For example, a shared random-effects model may specify a longitudinal density conditional on \(\boldsymbol b_i\), a survival contribution conditional on the same effects, and a distribution for those effects. Its posterior has the general form
Here \(Y_i\) denotes subject \(i\)'s longitudinal measurements, \(T_i\) the observed event or censoring time, and \(\delta_i\) the event indicator. The posterior depends on both the likelihood contributions and the prior \(p(\theta)\). The diagnostics in this tutorial assume the model and priors have been specified correctly, the posterior is proper, and the Markov chains are intended to sample that posterior. They do not establish that the model is scientifically adequate.
Chain-Specific Traces and Competing Modes
A trace plot displays sampled values in their iteration order. Inspect traces separately by chain for important parameters, especially association parameters, random-effect variances and correlations, and derived quantities that characterize the longitudinal-survival link. A chain that explores one narrow region while another explores a separated region may indicate competing modes, poor movement between modes, or both. These possibilities require investigation; the trace alone does not determine which explanation is correct.
Compare the parameter values and log posterior across chains, and inspect whether chains move between the apparent regions. If each chain remains near a different mode, combining all draws can produce a misleading summary: a pooled mean may lie in a low-density region, and a pooled interval may suggest a continuous range of plausible values where the posterior actually has a valley. Chain-specific summaries help reveal this problem. A convergence statistic can flag disagreement between chains, but a satisfactory value is not proof that all relevant modes have been found or that the posterior is well identified.
A mode can also be missed entirely if all chains start in the same region. Use dispersed, scientifically plausible starting values and examine whether different starts lead to the same regions and comparable posterior behavior. If modes are found, do not interpret the fraction of draws from each mode as its posterior probability when chains have not mixed between modes: that fraction may reflect initialization and run length rather than the posterior mass of each region.
Worked Example: Two Chains Remain in Different Regions
Suppose, in a hypothetical joint analysis, the association parameter \(\alpha\) links the current latent marker value to the event hazard. Two chains of 2,000 retained draws are started from different plausible values. Chain A stays near \(\alpha=0.8\), while Chain B stays near \(\alpha=-0.75\). Both traces fluctuate within their respective regions but show no transitions between them. The log posterior values sampled in the two regions overlap substantially.
Interpretation. The separated traces are evidence that the sampler has not explored the two regions as one connected trajectory. Similar log posterior values make it plausible that both regions deserve attention, but do not establish their relative posterior mass. A pooled average would be approximately \( (0.8-0.75)/2=0.025 \) if the two chains were given equal weight. That value is near neither region and may be unrepresentative of the posterior density.
Next steps. Examine other parameters and derived quantities to determine whether the regions differ in a substantive way or reflect a parameterization symmetry. Run additional dispersed chains and assess whether transitions occur. Compare the model likelihood and prior contributions in each region, and check that the model has been coded and parameterized as intended. Until the chains adequately represent the posterior regions and their relative mass, do not present the pooled mean \(0.025\) as a meaningful estimate of the association.
Not every pair of separated traces represents true multimodality. A chain can be trapped by difficult posterior geometry, and a parameter transformation can sometimes create equivalent representations of the same model. Conversely, chains that overlap in a few monitored parameters may still differ in unmonitored quantities. Check the joint pattern of important parameters rather than relying on one trace or a single scalar convergence summary.
Prior Sensitivity: Regularization or Information?
A prior sensitivity analysis asks whether conclusions change under a set of alternative, scientifically reasonable priors. Change the prior for the parameter of interest while keeping the likelihood, data, and other modeling choices fixed as far as possible. Then compare posterior locations, uncertainty, and any conclusions relevant to the scientific question. The goal is not to find a prior that produces a preferred answer; it is to assess how much the data constrain the result beyond the chosen prior.
When the likelihood is concentrated relative to the alternative priors, posterior summaries may be similar across those priors. This is evidence that the data provide substantial information in that parameter direction. When the likelihood is broad, different reasonable priors can yield noticeably different posterior summaries. That pattern indicates prior regularization: the prior is materially shaping the posterior. It does not mean the posterior is invalid. It does mean the result should not be described as strongly determined by the data alone.
A useful numerical illustration is a scalar parameter with an approximately normal likelihood and a normal prior. This approximation is for explaining the prior-likelihood balance; it does not assert that a joint model’s exact likelihood is normal.
In detail, the log of the likelihood-prior product, up to a constant, is
The coefficient of \(-\alpha^2/2\) is \(1/s_L^2+1/\tau^2\), the posterior precision. The coefficient of \(\alpha\) is \(\hat\alpha/s_L^2+\mu_0/\tau^2\). Completing the square therefore gives posterior variance \(v\) equal to the inverse precision and posterior mean \(m\) equal to the variance times that linear coefficient. The expression makes the key diagnostic idea explicit: posterior behavior reflects the relative precision of likelihood and prior.
Worked Example: A Weak Likelihood and a Regularizing Prior
Suppose the hypothetical likelihood for a joint-model association parameter is approximately \(N(0.40,2.25)\) as a function of \(\alpha\). Its standard error is \(1.5\), so the likelihood is broad. Compare priors \(N(0,1)\) and \(N(0,1/16)\). Both are centered at zero, but the second is much more concentrated.
With prior \(N(0,1/16)\), the prior precision is \(16\), and the likelihood precision is \(1/2.25=4/9\). Thus
The posterior standard deviation is \(\sqrt{9/148}\approx 0.2466\). Using a normal 95% posterior interval gives \(0.01081\pm1.96(0.2466)\), or approximately \([-0.4725,\,0.4941]\).
With prior \(N(0,1)\), the prior precision is \(1\). The posterior is
The posterior standard deviation is \(\sqrt{9/13}\approx 0.8321\), and the corresponding normal 95% posterior interval is \(0.1231\pm1.96(0.8321)\), or approximately \([-1.508,\,1.754]\). The substantial difference in posterior spread, along with the shift in center, shows that the prior choice materially affects this weakly informed parameter. These intervals describe posterior uncertainty under each specified prior and likelihood approximation; they are not frequentist confidence intervals.
Worked Example: A Concentrated Likelihood and Similar Posteriors
Now suppose a different hypothetical data set gives an approximate likelihood \(N(0.65,0.01)\) for \(\alpha\), with standard error \(0.10\). Compare priors \(N(0,1)\) and \(N(0,0.25)\). The likelihood precision is \(100\), much larger than either prior precision.
For \(N(0,1)\), the prior precision is \(1\), so \(v=1/(100+1)=1/101\approx0.009901\) and \(m=(1/101)(100\times0.65)=65/101\approx0.6436\). The posterior standard deviation is \(\sqrt{1/101}\approx0.09950\), giving the normal 95% posterior interval \([0.4485,\,0.8386]\).
For \(N(0,0.25)\), the prior precision is \(4\), so \(v=1/(100+4)=1/104\approx0.009615\) and \(m=(1/104)(100\times0.65)=65/104=0.6250\). The posterior standard deviation is \(\sqrt{1/104}\approx0.09806\), giving the interval \([0.4328,\,0.8172]\).
The posterior centers and spreads remain fairly similar despite the different prior variances. In this approximation, the likelihood dominates the prior precision, supporting the interpretation that these data provide substantial information about \(\alpha\). The check is conditional on the model and on the range of priors examined; it is not proof that the model is correct or that every plausible prior would give the same result.
Diagnostics, Alternatives, and Limits
A practical assessment combines chain behavior with prior sensitivity. First identify parameters for which substantive interpretation matters. Inspect separate traces and chain-specific summaries, especially for parameters whose values may lie in different modes. Then repeat the fit under alternative plausible priors and compare posterior summaries on the same scale. If a posterior changes greatly under modest prior changes, report that dependence rather than presenting one prior-specific result as a data-determined finding.
Different diagnostics answer different questions. Trace plots and between-chain comparisons can reveal that chains disagree, but cannot guarantee that all modes have been found. Posterior predictive checks assess whether the fitted model can reproduce relevant features of the observed data; good predictive fit does not establish that every parameter is identified. A likelihood-based profile or information analysis can provide a complementary view of data curvature, but it does not replace examining the posterior under the stated prior. These tools are complementary, not substitutes.
| Evidence | What it can indicate | What it cannot establish by itself |
|---|---|---|
| Separated chain-specific traces | Chains may occupy different posterior regions or fail to move between them | The true number of modes or their relative posterior mass |
| Similar posterior summaries under several reasonable priors | The data may dominate those prior choices for the summaries examined | Robustness to every possible prior or correctness of the model |
| Large posterior changes across reasonable priors | Material prior regularization or weak information in the likelihood | That the analysis is automatically unusable or that one prior is correct |
| Good posterior predictive agreement | The model can reproduce selected features of the data | Strong identification of each model parameter |
Prior sensitivity conclusions depend on which alternatives are considered. A narrow set of nearly identical priors can hide sensitivity, while an implausibly broad set can make differences look more alarming than they are. State the alternatives and why they are reasonable for the parameter’s scale and scientific meaning. Also check that prior changes do not inadvertently alter other aspects of the model, such as the implied distribution of random effects.
One limitation is that an apparently stable posterior under several priors can still reflect a mode that every chain missed. Another is that a parameter may be weakly identified while a particular combination of parameters is well estimated. Therefore, interpret parameter-specific diagnostics alongside meaningful functions of the parameters. Posterior multimodality and prior sensitivity are evidence about the behavior of the fitted model, not standalone tests of scientific truth.
Common Pitfall
Wrong reasoning: “The pooled chains give a plausible average, so the posterior must be well explored.” If separate chains remain in separated regions, the average can fall in a low-density gap, and their relative numbers of draws may be determined by starting points.
Correction: inspect chain-specific traces and summaries before pooling, investigate the regions and whether chains move between them, and do not report a pooled summary as representative until posterior exploration is adequate. In parallel, compare reasonable prior choices to determine whether the parameter’s posterior is strongly shaped by the prior. Neither a single convergence statistic nor a single prior fit answers both questions.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two chains for a joint-model association parameter remain in separate regions and do not transition between them. Why might their pooled mean be misleading?
- A likelihood approximation for \(\alpha\) is \(N(0.30,1)\), and the prior is \(N(0,1)\). Find the posterior mean and variance.
- In a prior sensitivity check, posterior intervals are nearly unchanged under several reasonable priors. What does this support, and what does it not prove?
- Why does a satisfactory convergence statistic not guarantee that a multimodal posterior has been fully explored?
Solutions. (1) The mean can lie between modes, in a low-density region, and the pooled fraction of draws in each region may reflect initialization rather than posterior mass. (2) Both precisions are \(1\), so \(v=1/(1+1)=0.5\) and \(m=0.5(0.30)=0.15\). (3) It supports the view that the data dominate those particular prior choices for the summaries examined; it does not prove robustness to all priors, that all modes were found, or that the model is correct. (4) A statistic summarizes selected features of sampled chains. If all chains miss the same mode, or chains inadequately explore separated regions, that statistic may fail to reveal the omitted posterior structure.