Bayesian Sequential and Joint Inference

Why Bayesian estimation strategy matters

A PK estimate is uncertain, even when it is reported as one number. If a PD analysis substitutes that number for the unknown PK parameter, the resulting PD posterior is conditional on the substitution. Its credible interval does not automatically include the uncertainty in the PK estimate. Bayesian analysis makes this distinction explicit: one can condition on a fixed PK value, carry forward a distribution of PK values, or estimate PK and PD parameters together from both data sources.

These approaches answer related but not identical questions. A plug-in analysis asks what the PD posterior would be if the selected PK estimate were known exactly. An uncertainty-propagating sequential analysis uses PK information to form a PK posterior, then passes that distribution to the PD analysis without allowing PD data to revise it. A joint analysis uses both PK and PD data to update the parameters together. The choice matters especially when exposure is uncertain, the PD model is nonlinear in exposure, or the PD data are informative about PK parameters.

Posterior factorization and three analyses

Let \(\theta\) denote PK parameters, \(\phi\) PD parameters, \(C\) the PK observations, and \(Y\) the PD observations. Write \(p(\theta)\) and \(p(\phi)\) for prior densities, \(p(C\mid\theta)\) for the PK data model, and \(p(Y\mid C,\theta,\phi)\) for the PD data model. The PD model may use concentrations predicted from \(\theta\), as well as \(\phi\). For clarity, first suppose the prior factors as \(p(\theta)p(\phi)\); a dependent joint prior must instead be represented explicitly as \(p(\theta,\phi)\).

Under the factorized prior and the stated likelihoods, the joint posterior is

A plug-in PD posterior at a fixed \(\theta^*\) is proportional to \(p(Y\mid C,\theta^*,\phi)p(\phi)\). For sequential inference, first compute \(p(\theta\mid C)\propto p(C\mid\theta)p(\theta)\). At each possible \(\theta\), the conditional PD posterior is proportional to \(p(Y\mid C,\theta,\phi)p(\phi)\). A cut or modular posterior combines these conditional PD posteriors with the PK-only posterior for \(\theta\), so the PD observations do not change the PK distribution. It is not generally the same distribution as the joint posterior: in the joint analysis, \(\theta\) is reweighted by how well each value predicts \(Y\).

This distinction also clarifies the role of prior information. A prior on \(\phi\) enters the PD posterior in all three approaches. In a sequential analysis, the PK prior contributes to \(p(\theta\mid C)\), which is then passed to the PD stage. In a joint analysis, both likelihood contributions and both priors enter once. Reusing the PK observations a second time after using \(p(\theta\mid C)\) as a distribution at the PD stage would count that information twice.

Worked Example: Plug-in versus propagated PK uncertainty

How joint inference adds feedback

The joint posterior can be understood as the PK-only posterior reweighted by the PD likelihood. In particular, the marginal posterior for \(\theta\) is proportional to \(p(\theta\mid C)\) multiplied by the PD marginal likelihood at that \(\theta\), where the PD parameters have been integrated out. Thus values of \(\theta\) that make the observed PD data more plausible can gain posterior weight. This feedback is appropriate when the shared model and priors are credible; it may be undesirable if the purpose is to protect PK estimates from a potentially misspecified PD model.

Prior information and modeling assumptions

A prior can materially affect PD inference when the response data provide limited information. To isolate that effect, keep the plug-in value \(\theta=10\) and the same centered response \(z=6\) with residual variance \(4\). If the prior changes from \(\phi\sim N(0,9)\) to \(\phi\sim N(0,1)\), the posterior variance becomes \((1+1/4)^{-1}=0.800\), and the posterior mean becomes \(0.800(6/4)=1.200\) response units. With the original prior, the mean was \(4.154\). This is not evidence that one prior is preferable; it shows why the prior must represent defensible external information and why sensitivity to plausible prior choices should be examined.

These calculations rely on correctly specified likelihoods, appropriately scaled parameters, and priors that encode the intended information. The normal examples also assume known residual variances and independent measurement errors. In an application, inspect posterior simulation diagnostics, compare simulated and observed concentrations and responses, and check whether posterior summaries are sensitive to reasonable alternative priors and model assumptions. A chain that has not mixed adequately does not provide trustworthy posterior summaries. Predictive discrepancies can reveal problems such as systematic residual patterns, but good diagnostics cannot prove the model is correct.

Derivation: When uncertainty propagation changes the variance

The preceding sequential example has a short general justification. Suppose the PK-only posterior for \(\theta\) has mean \(m\) and variance \(V_\theta\), and the PD response is \(Y=a+\phi+b(\theta-m)+\epsilon\), with \(\epsilon\sim N(0,\sigma^2)\). Conditional on \(\theta\), the centered observation \(y-a\) implies a normal likelihood for \(\phi\) with mean \(y-a-b(\theta-m)\) and variance \(\sigma^2\). If the prior for \(\phi\) is \(N(\mu_0,V_0)\), normal updating gives conditional posterior variance \(V=(1/V_0+1/\sigma^2)^{-1}\) and conditional mean \(V(\mu_0/V_0+[y-a-b(\theta-m)]/\sigma^2)\). The conditional mean has slope \(-bV/\sigma^2\) with respect to \(\theta\). Averaging its variance across the PK posterior therefore gives

The first term is the uncertainty conditional on a known PK value; the second is the variance of conditional PD posterior means across PK values. It is nonnegative, and it vanishes if \(b=0\) or \(V_\theta=0\). The result assumes the stated linear-normal model, independent priors, and a cut that retains the PK-only posterior for \(\theta\). It does not assert that a joint posterior must have a larger PD variance: joint feedback changes the PK distribution as well, and the resulting marginal variance depends on the full model.

Choosing and checking the analysis

The main practical distinction is whether the PD data should update PK parameters, not simply whether one method is “Bayesian.” Plug-in inference is computationally simple, but treats the selected PK value as known. Sequential propagation retains PK uncertainty but blocks feedback. Joint inference allows feedback and represents the full posterior under the specified joint model, but requires credible PK and PD likelihoods and priors together. If a PD model is nonlinear in predicted concentrations, the conditional PD posterior can vary substantially across PK posterior draws; substituting a mean concentration or mean PK parameter may then fail to represent the average posterior behavior.

A useful computational check for sequential propagation is to draw \(\theta^{(s)}\) from \(p(\theta\mid C)\), obtain the conditional PD posterior for each draw, and summarize the resulting mixture rather than reporting only a fit at the PK posterior mean. For joint inference, examine convergence and effective sampling of both PK and PD parameters, their posterior dependence, and posterior predictions for both data types. In either case, posterior intervals express uncertainty conditional on the model and prior assumptions; they do not account automatically for model misspecification.

Common pitfall: “The PK posterior mean is 10, so using 10 in the PD model preserves the PK uncertainty.” It does not. A single value has no spread, and the plug-in posterior interval is conditional on that value. To carry uncertainty forward, integrate or average over the PK posterior; to allow PD data to revise PK estimates, use the joint posterior. These are distinct operations.