Posterior Predictive Distributions for Future Concentrations

Why a Future Concentration Is a Distribution

After observing an individual's concentration data, the individual's pharmacokinetic parameters are still uncertain. A future concentration therefore has two sources of variability: uncertainty about the individual's parameters, and variability in the future concentration conditional on those parameters. A forecast that accounts for both is a distribution, not just the concentration calculated from one selected parameter estimate.

Let $D_i$ denote the observed data for individual $i$, $\boldsymbol{\theta}_i$ the individual's parameter vector, and $\boldsymbol{\psi}$ the population-model parameters. Let $C_i^*$ be a future observed concentration at a specified time and under a specified dosing history. The conditioning information, including that time and dosing history, is held fixed throughout. The posterior distribution $p(\boldsymbol{\theta}_i\mid D_i,\boldsymbol{\psi})$ represents uncertainty about the individual parameters after observing $D_i$.

Definition: The posterior predictive distribution of a future concentration is the conditional distribution of $C_i^*$ given the observed individual data and the population-model parameters. Under conditional independence of future and observed observations given $\boldsymbol{\theta}_i$ and $\boldsymbol{\psi}$, it is $$ p(C_i^*\mid D_i,\boldsymbol{\psi}) = \int p(C_i^*\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})\, p(\boldsymbol{\theta}_i\mid D_i,\boldsymbol{\psi})\, d\boldsymbol{\theta}_i . $$ The first factor describes future concentration variability at fixed individual parameters; the second weights parameter values by their posterior plausibility.

The formula is an average of distributions, not an instruction to average the parameters first. It applies whether the posterior is represented by a continuous density or by a discrete set of parameter values and weights. If $\boldsymbol{\psi}$ is estimated and then held fixed, the result is conditional on that estimate; it does not include uncertainty in the population parameters.

Deriving the Predictive Distribution

The conditional-independence assumption in the definition means that, once the individual's parameters and population model are specified, the future observation provides no additional information about the already observed data. In symbols, $C_i^*$ is conditionally independent of $D_i$ given $\boldsymbol{\theta}_i,\boldsymbol{\psi}$. The model then factors the joint conditional distribution as $p(C_i^*,D_i\mid\boldsymbol{\theta}_i,\boldsymbol{\psi}) =p(C_i^*\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})p(D_i\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})$.

To obtain a distribution for $C_i^*$ that reflects uncertainty about $\boldsymbol{\theta}_i$, average its conditional density over the posterior. This follows directly by conditioning on the possible values of $\boldsymbol{\theta}_i$, then integrating those values out:

\begin{aligned} p(C_i^*\mid D_i,\boldsymbol{\psi}) &=\int p(C_i^*,\boldsymbol{\theta}_i\mid D_i,\boldsymbol{\psi})\, d\boldsymbol{\theta}_i\\ &=\int p(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})\, p(\boldsymbol{\theta}_i\mid D_i,\boldsymbol{\psi})\, d\boldsymbol{\theta}_i\\ &=\int p(C_i^*\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})\, p(\boldsymbol{\theta}_i\mid D_i,\boldsymbol{\psi})\, d\boldsymbol{\theta}_i . \end{aligned}

The last equality uses the conditional-independence assumption; it is not automatic. If future and observed residuals are correlated, that equality may fail. The general identity instead uses the conditional distribution given both the individual parameters and the observed data: $p(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})$.

The same conditioning argument gives a useful variance decomposition. Define $m(\boldsymbol{\theta}_i,D_i)=E(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})$ and $v(\boldsymbol{\theta}_i,D_i)=\operatorname{Var}(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})$. When these moments exist, the law of total variance gives

\operatorname{Var}(C_i^*\mid D_i,\boldsymbol{\psi}) = E\{v(\boldsymbol{\theta}_i,D_i)\mid D_i,\boldsymbol{\psi}\} + \operatorname{Var}\{m(\boldsymbol{\theta}_i,D_i)\mid D_i,\boldsymbol{\psi}\}.

The first term averages the conditional future variance; the second is variation in the conditional mean across plausible individual parameters. Under conditional independence, $m$ and $v$ can be written as functions of $\boldsymbol{\theta}_i$ alone. Without it, they must retain their dependence on $D_i$.

Worked Example: A Two-Point Posterior and a Mixture Forecast

Suppose a hypothetical individual's posterior assigns probability 0.25 to one parameter value and 0.75 to another. At the planned future sampling time, the model gives $C_i^*\mid\boldsymbol{\theta}_i\sim N(8,2.25)$ mg/L for the first value and $C_i^*\mid\boldsymbol{\theta}_i\sim N(4,2.25)$ mg/L for the second. Assume future residuals are conditionally independent of the observed data. The numbers are hypothetical and the second argument of $N$ is the variance.

The posterior predictive distribution is the weighted mixture

p(C_i^*\mid D_i,\boldsymbol{\psi}) =0.25\,N(8,2.25)+0.75\,N(4,2.25).

Its mean is $0.25(8)+0.75(4)=5.00$ mg/L. Its variance is the average conditional variance plus the variance of the conditional means: $0.25(2.25)+0.75(2.25)+0.25(8-5)^2+0.75(4-5)^2 =2.25+3.00=5.25\;(\text{mg/L})^2$. Thus its standard deviation is $\sqrt{5.25}=2.291$ mg/L.

If the more probable parameter value is also the empirical Bayes estimate (the posterior mode in this example), a plug-in forecast is $N(4,2.25)$. It has mean 4 mg/L and standard deviation 1.500 mg/L. The mixture forecast differs because it includes the less probable parameter value and its higher predicted concentration. The plug-in distribution is not the posterior predictive distribution.

Why Averaging Parameters Is Not Enough

A common shortcut is to calculate the concentration at a posterior mean parameter. In a nonlinear pharmacokinetic model, the concentration at the average parameter generally differs from the average concentration over the parameter posterior. Even if the future concentration is deterministic at fixed parameters, the posterior predictive distribution can still have variation because the parameters are uncertain.

Worked Example: Nonlinear Concentration at an Average Parameter

Consider a hypothetical one-compartment bolus model with fixed volume $V=10$ L and dose $100$ mg. At $t^*=5$ h, the concentration is $C_i^*= (100/10)\exp(-CL_i t^*/V)$ mg/L. Suppose the posterior assigns equal probability to $CL_i=1$ L/h and $CL_i=3$ L/h. Ignore residual error here to isolate parameter uncertainty.

At $CL_i=1$ L/h, the predicted concentration is $10\exp[-(1)(5)/10]=10e^{-0.5}=6.065$ mg/L. At $CL_i=3$ L/h, it is $10\exp[-(3)(5)/10]=10e^{-1.5}=2.231$ mg/L. The posterior predictive distribution puts probability 0.5 on each of these concentrations, so its mean is $0.5(6.065)+0.5(2.231)=4.148$ mg/L.

The posterior mean clearance is $0.5(1)+0.5(3)=2$ L/h. Evaluating the concentration model there gives $10e^{-1}=3.679$ mg/L, not 4.148 mg/L. Thus prediction at the posterior mean parameter is not generally the posterior predictive mean. This difference arises from the nonlinear exponential relationship, not from an arithmetic error.

When Observed and Future Residuals Are Dependent

The usual mixture formula assumes the residual for the future observation is conditionally independent of the residuals in $D_i$, given the individual parameters and population model. A residual process with serial correlation, for example, can violate this assumption. In that situation, the observed residuals help predict the future residual even when $\boldsymbol{\theta}_i$ is fixed. The appropriate predictive mixture must use $p(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})$, rather than $p(C_i^*\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})$.

Worked Example: A Correlated Future Residual

Suppose a hypothetical model has a fixed conditional mean concentration of 10 mg/L and residuals $(\varepsilon,\varepsilon^*)$ that are jointly normal, each with standard deviation 2 mg/L and correlation $\rho=0.5$. The observed concentration is 12 mg/L, so the observed residual is 2 mg/L, or one residual standard deviation above its mean. There is no parameter uncertainty in this example.

For jointly normal residuals, conditioning on the observed residual gives $E(\varepsilon^*\mid\varepsilon)=\rho\varepsilon$ and $\operatorname{Var}(\varepsilon^*\mid\varepsilon)=\sigma^2(1-\rho^2)$. Substituting $\varepsilon=2$, $\rho=0.5$, and $\sigma=2$ mg/L gives a future residual mean of $0.5(2)=1$ mg/L and variance $2^2(1-0.5^2)=3.00\;(\text{mg/L})^2$.

Consequently, the future concentration conditional on the observed data has mean $10+1=11$ mg/L and variance 3.00 $(\text{mg/L})^2$. If one incorrectly used the unconditional residual distribution at fixed parameters, the forecast would have mean 10 mg/L and variance 4.00 $(\text{mg/L})^2$. The difference occurs even though the individual parameter is known exactly.

Practical Checks and Limits

A posterior predictive forecast is only as credible as its posterior and its conditional future-observation model. For each forecast, make the target explicit: the future time, dosing history, and whether the target is a latent model concentration or a future measured concentration. A measured concentration ordinarily includes residual variability; a latent concentration does not. Mixing those targets changes the predictive distribution.

Several checks help reveal problems in the calculation or its assumptions:

Check the conditioning. Confirm that the individual posterior incorporates the observed data and that the future concentration is evaluated under the intended future regimen and time.
Check the residual structure. If residuals are correlated over time, or share a measurement-level process, do not assume conditional independence without justification. Model the joint dependence and condition the future residual on the observed information.
Check parameter support and model behavior. Posterior values should be physiologically plausible under the model, and predictions across that support should not produce unexplained discontinuities or impossible values.
Check mixture construction. Every posterior weight must be nonnegative and the weights must sum to one. The conditional distributions must use the same target quantity and units.
Check whether population parameters are treated as fixed. Conditioning on $\boldsymbol{\psi}$ omits uncertainty in those parameters. A forecast intended to include it needs an additional averaging step over uncertainty in $\boldsymbol{\psi}$.

The main alternative is a plug-in forecast at an empirical Bayes estimate, such as a posterior mode. It is simpler and may be useful as a single representative prediction, but it suppresses variation across the individual posterior. Evaluating at a posterior mean parameter is another plug-in choice; in nonlinear models it need not equal even the predictive mean. Neither single-parameter approach captures the full posterior predictive distribution. Conversely, averaging over a posterior cannot correct a misspecified structural or residual model: it propagates uncertainty under the assumed model, not uncertainty about whether that model is right.

Key takeaway: The posterior predictive distribution averages the future concentration distribution over plausible individual parameters. Use the ordinary mixture formula only when future and observed data are conditionally independent given those parameters and the population model; otherwise condition the future distribution on the observed data as well.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In the posterior predictive integral, what does each of the two density factors represent?
A posterior assigns probabilities 0.6 and 0.4 to two parameter values. Their conditional future concentration means are 5 and 9 mg/L, respectively, and both conditional variances are 1 $(\text{mg/L})^2$. Find the predictive mean and variance.
Why can concentration evaluated at a posterior mean clearance differ from the posterior predictive mean concentration?
When is it invalid to replace $p(C_i^*\mid\boldsymbol{\theta}_i,D_i,\boldsymbol{\psi})$ with $p(C_i^*\mid\boldsymbol{\theta}_i,\boldsymbol{\psi})$?

Solutions. (1) The future conditional density describes concentration variability at fixed individual parameters; the posterior density weights parameter values by their plausibility after observing $D_i$. (2) The mean is $0.6(5)+0.4(9)=6.6$ mg/L. The variance is the average conditional variance, 1, plus the variance of the conditional means: $0.6(5-6.6)^2+0.4(9-6.6)^2=3.84$. The total is 4.84 $(\text{mg/L})^2$. (3) The concentration model is generally nonlinear in clearance, so evaluating it at an average clearance does not in general equal averaging its values over the clearance posterior. (4) The replacement is invalid when observed and future concentrations are dependent after conditioning on the individual parameters and population model, as with correlated residuals.