The MCMC Target in Bayesian Joint Models

What Distribution Does MCMC Target?

In a Bayesian joint model, MCMC is used to generate draws from a posterior distribution. To understand what those draws represent, it is essential to identify the target precisely: it includes both population-level model parameters and the subject-specific random effects treated as latent quantities. The target is not just the longitudinal likelihood, just the survival likelihood, or a list of fitted estimates. It is a probability distribution formed by combining the two data sources with the model’s prior distributions.

Let \(i=1,\ldots,n\) index subjects, and let \(j=1,\ldots,n_i\) index the longitudinal measurements for subject \(i\). Write \(y_{ij}\) for the measurement at time \(t_{ij}\). Let \(T_i\) be the observed event or follow-up time and \(\delta_i\) indicate whether an event was observed. Let \(b_i\) be subject \(i\)’s random-effects vector. Denote the collection of global model parameters by \(\theta\), including, as appropriate, longitudinal fixed effects and residual parameters, the random-effects covariance matrix, survival regression and baseline hazard parameters, and association parameters.

Here \(\mathcal D\) denotes all observed data, \(L_{Yi}\) is subject \(i\)’s conditional longitudinal likelihood, and \(L_{Ti}\) is that subject’s conditional survival likelihood. The proportionality omits the marginal data density, which is a constant with respect to \(\theta\) and \(b\). The prior factorization shown is a common specification: global parameters have prior density \(p(\theta)\), and subject effects are conditionally independent given the relevant parameters. If a model specifies a different joint prior, that prior replaces these factors; the likelihood construction is unchanged.

The elements of \(\theta\) are global or population-level unknowns. In contrast, \(b_i\) describes subject \(i\)’s deviation from the population trajectory or hazard structure. A random effect is a latent variable in the probability model, not an additional observed measurement. It is unknown and receives a posterior distribution, but it is not interchangeable with a population parameter such as a fixed-effect coefficient or an association parameter.

How the Two Likelihoods Enter

For a standard Gaussian longitudinal submodel with conditionally independent measurement errors, suppose \(y_{ij}\mid b_i,\theta\sim N\{m_i(t_{ij};b_i),\sigma^2\}\), where \(m_i(t;b_i)=x_i(t)^\mathsf T\beta+z_i(t)^\mathsf T b_i\). Then

For illustration, take a proportional-hazards specification in which the current longitudinal value enters the hazard: \(h_i(t\mid b_i,\theta)=h_0(t)\exp\{\gamma^\mathsf T w_i+\alpha m_i(t;b_i)\}\). With the usual event-time likelihood contribution, the survival factor is

This form assumes that the longitudinal observations are conditionally independent given the random effects and model parameters, and that the event-time process has the specified hazard conditional on those same quantities and covariates. It also relies on the model’s censoring assumptions, including that the event-time contribution is appropriate for the observed follow-up. These assumptions should be checked against the study design and model specification; they are not consequences of using MCMC.

The shared \(b_i\) is the source of the coupling. It appears in the longitudinal mean and, through \(m_i(t;b_i)\), in the hazard. Consequently, longitudinal observations inform the posterior for \(b_i\), and that posterior information affects the survival likelihood; survival information also informs \(b_i\), and thereby the implied longitudinal trajectory. Conditional on \(b_i\) and the model parameters, the two likelihood factors can be multiplied. They are not generally independent after \(b_i\) is integrated out.

Why the Posterior Is a Joint Target

The target follows directly from Bayes’ rule and the conditional likelihood factorization. Given \(\theta\) and \(b\), the model assigns the observed data a likelihood that factors by subject; within each subject it separates into longitudinal and survival contributions under the stated conditional-independence assumptions. Multiplying this likelihood by the prior density gives the posterior up to its normalizing constant:

Since \(p(\mathcal D)\) does not vary across candidate values of \(\theta\) and \(b\), it can be omitted when specifying the MCMC target. It cannot be omitted from a calculation that requires the normalized posterior density itself. MCMC targets the distribution represented by the resulting kernel, not a particular numerical value of that kernel.

This also explains why a subject effect should not be silently dropped from the target. If \(b_i\) is represented explicitly, its prior density and both conditional likelihood contributions belong in the joint posterior. A posterior for \(\theta\) alone would instead require integrating the subject effects out: \(p(\theta\mid\mathcal D)=\int p(\theta,b\mid\mathcal D)\,db\). That is a different representation of the posterior, not the same explicit joint target with the \(b_i\) terms omitted.

Assumptions, Diagnostics, and Alternatives

Writing down the target is also a useful model check. Before interpreting its terms, verify that the longitudinal density matches the measurement scale and error structure, the hazard uses the intended association structure, the event-time contribution matches the observation and censoring scheme, and all prior factors correspond to the parameters in the model. Check that a random effect used in one component is the same subject’s effect in the other component. If the longitudinal and survival submodels use unrelated subject effects, the target no longer expresses the shared-random-effects model described here.

Several warning signs indicate that the target may have been assembled incorrectly:

• A missing prior factor. A latent \(b_i\) included in the sampling state still needs its random-effects density in the posterior kernel.
• A missing data component. Using only \(L_{Yi}\) or only \(L_{Ti}\) discards information and does not target the stated joint model.
• Duplicated contributions. Each subject’s longitudinal and survival likelihood should enter once under the stated factorization, not once per MCMC iteration within the mathematical target.
• Inconsistent dependence. If the survival hazard uses \(m_i(t;b_i)\), but the likelihood calculation treats that term as independent of \(b_i\), the shared-effect coupling has been lost.
• Confusing kernel and density. Omitting a parameter-independent normalizing constant is valid for the MCMC target kernel, but it does not make the kernel a normalized posterior density.

A basic diagnostic is to inspect the written target term by term and check it at a fixed, simple parameter value, as in the worked calculations. Also verify that changing \(b_i\) can affect the appropriate longitudinal and survival factors, and that changing a global parameter affects precisely the components in which it appears. These checks can find coding or model-translation errors; they do not establish that a sampling algorithm has converged or that the model is substantively adequate.

The explicit joint target with \(b\) included is one computational representation. An alternative is to integrate the random effects out and target a marginal posterior for \(\theta\); this can avoid sampling those latent variables explicitly but requires evaluating the resulting marginal likelihood. Another alternative is to fit longitudinal and survival models separately, but separate fits generally do not use the shared random effect to transmit information between the two processes. Neither alternative changes the model automatically: the likelihood and prior must still encode the intended assumptions.

A common pitfall is to say, “MCMC samples the parameter posterior, so the random effects are only nuisance estimates and can be left out.” That reasoning confuses the marginal posterior for global parameters with the joint posterior used by an explicit latent-variable representation. If the algorithm’s state includes \(b\), the target must include \(p(b\mid\theta)\), \(L_Y(\theta,b)\), and \(L_T(\theta,b)\). A later marginal posterior for \(\theta\) can be obtained by integrating the joint posterior over \(b\), not by deleting its factors.