Bayesian Predictions From Posterior Draws

From Posterior Draws to a Dynamic Risk

At a landmark time \(s\), a patient is known to be alive and their longitudinal measurements through \(s\) are available. The prediction of interest is often the probability of an event during the next \(w\) time units, conditional on that history and survival to \(s\). In a Bayesian joint model, this probability depends on uncertain population parameters and on the patient’s latent effects, which determine their underlying marker trajectory. Posterior draws provide a direct way to carry both sources of uncertainty into the prediction.

The key operation is to calculate the conditional risk for each joint draw, then average those risks. It is generally not equivalent to first average the model parameters and latent effects and then calculate one risk. The risk function is nonlinear, and the posterior dependence between parameters and patient effects is part of the prediction.

Write the event hazard conditional on a parameter value and latent effects as \(h_i(t\mid\theta,b_i)\). This hazard may depend on the latent marker trajectory, for example through a current-value association. The draw-specific risk is obtained from the cumulative hazard between \(s\) and \(s+w\). In applying the formulas below, assume the joint model specifies this hazard, the patient’s event history is correctly represented, and any future covariates needed for the hazard are fixed or otherwise specified for the prediction. This tutorial does not infer an unknown future covariate path.

Here \((\theta^{(m)},b_i^{(m)})\) is the \(m\)th joint posterior draw. The draw-specific risk is conditional on that draw’s parameters and latent effects. If the posterior draws target the distribution given the patient’s longitudinal history and survival to the landmark, the posterior predictive risk is estimated by

This is a Monte Carlo approximation to the posterior expectation of the draw-specific risk. The \(M\) draws may come from a Markov chain; in that case, they need not be independent. The average remains the basic estimator, but autocorrelation affects its Monte Carlo precision.

Why the Landmark Conditioning Matters

For fixed \(\theta\) and \(b_i\), the conditional survival probability from \(s\) through \(s+w\) is the ratio of survival probabilities at the two times. Under the hazard model, that ratio is the exponential of minus the cumulative hazard over the interval. Thus the conditional event risk for fixed latent quantities is one minus this survival probability.

Averaging this conditional probability over the posterior distribution given \(\mathcal{D}_i(s)\) and \(T_i>s\) gives the prediction. This conditioning is consequential: survival to the landmark is information about the patient’s latent risk and model parameters. If posterior draws are generated for a patient’s prediction, their target must incorporate the relevant event-free survival information up to \(s\), as well as the available longitudinal history.

The formula also explains the computational procedure: evaluate the bracketed risk at each posterior draw, then average. It is the risk—not the hazard, the marker value, or the parameters—that is averaged. The integral over time may be analytic for a simple hazard model or evaluated numerically when the hazard has a more complicated time dependence.

Worked Example: Constant Hazard Within Each Draw

Retaining Joint Parameter and Trajectory Uncertainty

The latent marker value is usually itself calculated from the longitudinal submodel and its patient-specific effects. For instance, a draw might imply \(m_i(s)=x_i(s)^\mathsf{T}\beta+z_i(s)^\mathsf{T}b_i\), where \(\beta\) and \(b_i\) are drawn together with the event-model parameters. A prediction must use the marker value implied by the same draw as the hazard parameters. Pairing is important because posterior draws can exhibit dependence: a larger latent marker level may be associated with a different value of the marker-event association parameter or other model parameters.

This calculation includes uncertainty in both the event-model parameters and the patient’s latent marker level, provided both are represented in the posterior draws. It does not mean that the marker’s measurement error is added directly to the event hazard as though an observed measurement were the true marker. Rather, uncertainty about the latent trajectory is represented through its posterior distribution conditional on the observed history and the fitted joint model.

Posterior Averaging Is Not a Plug-In Calculation

A plug-in prediction substitutes a single value, such as a posterior mean, for each unknown quantity and calculates one risk. Posterior-draw prediction instead calculates a risk at each draw and averages. Since exponentiation and the conversion from cumulative hazard to risk are nonlinear, these two operations generally differ. A plug-in calculation may be a useful quick summary, but it is not, in general, the Bayesian posterior predictive mean risk.

Implementation Checks and Failure Modes

Before interpreting a prediction, check both the posterior computation and the risk calculation. Posterior draws should come from a well-mixed chain targeting the intended conditional posterior; simply having a large number of saved iterations does not ensure a large effective sample size. Inspect trace plots and standard convergence diagnostics for the model parameters and patient-specific effects that affect the prediction. For correlated draws, assess Monte Carlo error using an appropriate method such as batch means or an effective-sample-size-based calculation, rather than treating all iterations as independent.

For each draw, verify that the cumulative hazard is nonnegative and that the resulting risk lies between zero and one. If numerical integration is used, check that its time grid is sufficiently fine or compare it with a more accurate integration method. Confirm the prediction horizon and time units: a baseline hazard expressed per year must be integrated over years, not days. Also verify that the patient’s marker value or trajectory is evaluated at the times used by the event model.

A frequent failure is to obtain draws conditional on the longitudinal observations but omit the fact that the patient survived to the landmark. That targets the wrong posterior distribution for a landmark prediction. Another is to calculate one risk from posterior mean parameters and describe it as the posterior predictive risk. A third is to average hazards first and then convert the average hazard to risk; that is generally not equal to averaging draw-specific risks. In all cases, the correct target is an average of risks evaluated under draws from the appropriate joint posterior.

The earlier material on Monte Carlo integration for risk predictions provides the general numerical-integration perspective. The Bayesian version here uses posterior draws to perform the integration over uncertain parameters and latent effects together. It is flexible, but it does not repair a poorly specified joint model or a chain that has not explored its target distribution. Prediction results remain conditional on the model assumptions, the observed history, and the treatment of future covariates.