Deriving the Individual Posterior for a PK Model

Why the Individual Posterior Has Two Parts

An individual PK forecast combines two sources of information. The population model, conditioned on the patient’s covariates, describes plausible values of that patient’s PK parameters before using the patient’s current concentration measurements. The patient’s dosing and concentration data then provide evidence about which parameter values are more compatible with the observed record. The individual posterior is the distribution that combines these sources.

The multiplication is simple to write, but its meaning depends on defining the prior, the likelihood, and the normalizing constant correctly. The likelihood must represent the concentration data under the stated PK and observation models; the prior must represent the population-derived distribution for this patient; and the normalizer must account for all parameter values over the prior’s support.

Definition: Let $\theta_i$ denote the vector of PK parameters for patient $i$, $x_i$ the patient’s covariates, $\psi$ the population-model quantities treated as fixed, $\mathbf{y}_i$ the patient’s observed concentrations, and $\mathcal{E}_i$ the known dosing and sampling design. The individual posterior density is $$ p(\theta_i\mid \mathbf{y}_i,\mathcal{E}_i,x_i,\psi) = \frac{ p(\mathbf{y}_i\mid \theta_i,\mathcal{E}_i)\, p(\theta_i\mid x_i,\psi) }{ m(\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi) }. $$ The prior $p(\theta_i\mid x_i,\psi)$ is the population-derived individual prior, the likelihood $p(\mathbf{y}_i\mid \theta_i,\mathcal{E}_i)$ describes the observed concentrations conditional on the individual parameters and design, and $m(\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi)$ is the marginal likelihood, also called the evidence.

For a continuous parameter vector, the evidence is the integral of the likelihood times the prior over the parameter space $\Theta$:

m(\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi) = \int_{\Theta} p(\mathbf{y}_i\mid \theta_i,\mathcal{E}_i) p(\theta_i\mid x_i,\psi)\,d\theta_i.

This integral is a density for the observed data, conditional on the design and covariates. It is not a density over $\theta_i$. For a discrete parameter space, the integral is replaced by a sum over its possible values. Provided the prior is a proper probability distribution and the evidence is finite and positive, dividing by the evidence makes the posterior integrate to one. Since the evidence does not depend on $\theta_i$, the posterior is sometimes written as proportional to the product of likelihood and prior; that shorthand does not remove the need for normalization when a posterior density or probability is required.

Deriving the Product and Its Normalizer

The result follows by expressing the joint distribution of the parameters and data in two ways. Given the patient’s covariates and the population-model quantities, the joint density of individual parameters and observed concentrations is the prior times the conditional data density:

p(\theta_i,\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi) = p(\mathbf{y}_i\mid \theta_i,\mathcal{E}_i)\, p(\theta_i\mid x_i,\psi).

The same joint density is the marginal density of the observed data times the conditional density of parameters given those data:

p(\theta_i,\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi) = p(\theta_i\mid \mathbf{y}_i,\mathcal{E}_i,x_i,\psi)\, p(\mathbf{y}_i\mid \mathcal{E}_i,x_i,\psi).

Equating the expressions and dividing by the marginal data density gives Bayes’ rule. Integrating the resulting posterior over $\theta_i$ gives one. Therefore the denominator must equal the integral of the numerator over all possible individual parameters. This also shows why the evidence is not an arbitrary scaling choice: it is exactly the quantity that normalizes the conditional distribution.

The population-model parameters $\psi$ are treated as fixed in this expression. If their uncertainty is also to be represented, a different, expanded model must account for it. Likewise, $\mathcal{E}_i$ is conditioned on: the dose history and sampling times are inputs to the concentration model, not additional observations to multiply into the concentration likelihood unless a separate probability model is specified for them.

Assumptions Behind the Construction

The posterior formula is general, but a useful posterior requires a defensible prior and likelihood. The earlier material on population PK models as individual parameter priors and conditioning individual priors on patient covariates motivates the prior term. The likelihood must describe how concentrations arise under a specified PK model and observation model. In applying the product, check the following:

Consistent parameterization: the prior and likelihood must refer to the same $\theta_i$, with compatible parameter units and support.
Relevant patient information: covariates used to form the prior should be those available for this patient, and the conditioning should match the stated population model.
Known design, unless modeled otherwise: the dose and sampling record used in the likelihood must be appropriate to the patient. Uncertain dose records cannot be silently treated as known without qualification.
Appropriate observation model: the likelihood must account for the stated residual error and any dependence among measurements. A product of separate observation densities requires conditional independence given the individual parameters and design.
Well-defined normalization: the prior must be integrable, and the likelihood-prior product must have a finite, nonzero integral.

For conditionally independent measurements, the likelihood factors as

p(\mathbf{y}_i\mid \theta_i,\mathcal{E}_i) = \prod_{j=1}^{n_i} p(y_{ij}\mid \theta_i,\mathcal{E}_i),

where $n_i$ is the number of measurements. This factorization is an assumption about the observation model, not a consequence of observing several concentrations. If residual errors are correlated, the joint likelihood must represent that dependence instead of multiplying marginal densities as though they were independent.

Worked Example: One Concentration Updates a Gaussian Prior

Consider a hypothetical local concentration model for one patient. Let $\theta$, measured in mg/L, be an individual additive deviation from a reference PK prediction. At the sampling time, the reference prediction is 10 mg/L, so the observed concentration is modeled as $Y=10+\theta+\varepsilon$. Before observing this patient’s concentration, suppose the population-derived conditional prior is $\theta\sim N(0,4)$, where 4 is the variance in $(\mathrm{mg/L})^2$. Suppose the measurement error is independent of $\theta$ and $\varepsilon\sim N(0,4)$. A hypothetical observation is $y=14$ mg/L. These assumptions make the residual $r=y-10=4$ mg/L normally distributed around $\theta$ with variance 4.

The likelihood as a function of $\theta$ is proportional to the density of $r=4$ under $N(\theta,4)$, and the prior is $N(0,4)$. Combining their precisions gives posterior variance

V_{\text{post}} = \left(\frac{1}{4}+\frac{1}{4}\right)^{-1} =2\;(\mathrm{mg/L})^2.

The posterior mean is the posterior variance times the prior-precision-weighted prior mean plus the likelihood-precision-weighted observed residual:

\mu_{\text{post}} = 2\left(\frac{0}{4}+\frac{4}{4}\right) =2\;\mathrm{mg/L}.

Thus $\theta\mid y\sim N(2,2)$, with mean 2 mg/L and variance 2 $(\mathrm{mg/L})^2$. The evidence can be calculated by integrating the likelihood over the prior. Equivalently, under this model $Y$ has marginal distribution $N(10,4+4)$, so

p(y=14) = \frac{1}{\sqrt{2\pi(8)}} \exp\left(-\frac{(14-10)^2}{2(8)}\right) \approx 0.0519\;(\mathrm{mg/L})^{-1}.

This evidence is the normalizing density for the observed concentration. The calculation assumes the stated Gaussian prior, known reference prediction, and independent Gaussian measurement error; it is a hypothetical illustration, not a claim that all PK likelihoods have this form.

Worked Example: Several Conditionally Independent Concentrations

Suppose a hypothetical patient has three concentration observations. After subtracting the reference PK predictions at their respective sampling times, the residuals are $r_1=2$, $r_2=4$, and $r_3=3$ mg/L. Use the same additive parameter $\theta$, but suppose the prior is $\theta\sim N(0,9)$ and each residual has independent error with standard deviation 2 mg/L. Conditional independence is part of this example’s model assumption.

Each residual contributes precision $1/4\;(\mathrm{mg/L})^{-2}$, so the total posterior precision is the prior precision plus three likelihood contributions:

\frac{1}{V_{\text{post}}} = \frac{1}{9}+\frac{3}{4} = \frac{31}{36}\;(\mathrm{mg/L})^{-2}, \qquad V_{\text{post}}=\frac{36}{31}\approx1.161\;(\mathrm{mg/L})^2.

The weighted sum of residuals is $(2+4+3)/4=9/4\;(\mathrm{mg/L})^{-1}$. Therefore,

\mu_{\text{post}} = \frac{36}{31}\left(\frac{9}{4}\right) = \frac{81}{31} \approx 2.613\;\mathrm{mg/L}.

The normalized posterior is $N(81/31,36/31)$, with the mean in mg/L and variance in $(\mathrm{mg/L})^2$. Multiplying the three observation densities is justified here only because the example specifies independent errors conditional on $\theta$. If the measurements shared correlated residual error, their joint likelihood would not be this product, and the posterior calculation would change.

Worked Example: The Normalizer on a Finite Parameter Grid

A finite grid can make the role of the evidence visible without requiring an integral. Consider a hypothetical two-value approximation for an individual parameter, with values $\theta_A$ and $\theta_B$. Let their prior probabilities be 0.7 and 0.3. Suppose two observed categorical concentration patterns have conditional probabilities 0.2 and 0.5 under $\theta_A$, and 0.6 and 0.4 under $\theta_B$. Assume the patterns are conditionally independent given the parameter.

The unnormalized posterior weights are the prior probabilities multiplied by the joint likelihoods:

w_A=0.7(0.2)(0.5)=0.070, \qquad w_B=0.3(0.6)(0.4)=0.072.

The evidence is the sum of these weights, $m=0.070+0.072=0.142$. Dividing each weight by this normalizer gives

P(\theta_A\mid \text{data})=\frac{0.070}{0.142}\approx0.493, \qquad P(\theta_B\mid \text{data})=\frac{0.072}{0.142}\approx0.507.

The posterior probabilities sum to one. The data have shifted the initial probabilities, but the posterior still reflects both the likelihood and the prior. This finite calculation is a direct illustration of the sum version of the evidence; it does not imply that a continuous individual PK parameter should be treated as having only two possible values.

Diagnostics, Comparisons, and Limitations

A posterior calculation can be algebraically normalized and still be scientifically misleading if its ingredients do not match the patient or the data. Before interpreting the result, check that the prior has support where the likelihood is informative, that the recorded dose and sampling times are correctly represented, and that the residual model is plausible for the concentration pattern. If multiple concentrations are treated as independent, verify that this follows from the assumed observation model rather than convenience. A posterior that barely changes from the prior may indicate that the data are weakly informative under the model; a very large change can reflect strong data information, prior-data conflict, or a data/model problem. The formula alone cannot distinguish these explanations.

The posterior is not the same object as the likelihood. The likelihood compares parameter values using the observed data but does not by itself encode the population-derived prior. Nor is the posterior a point estimate: it is a normalized distribution over individual parameters. Later methods can summarize or approximate that distribution, but those summaries do not replace its definition. In particular, dropping the evidence is acceptable when writing a posterior kernel for calculations in which a common proportionality constant cancels; it is not acceptable when claiming to have a normalized density or posterior probabilities.

The construction also has limits. It conditions on the specified population-model quantities and covariates, and so does not automatically represent uncertainty in those inputs. It relies on the selected PK and observation models, and it can be sensitive to misspecification, sparse data, or a prior that is inappropriate for the patient. A normalizing constant may be difficult to calculate for a complex nonlinear model even though its mathematical definition is straightforward. In such cases, an approximation to the posterior must still target the same likelihood-times-prior distribution.

Common pitfall: “The likelihood is already the posterior, so I can just multiply the data densities and stop.” This omits the patient-specific prior and usually leaves an unnormalized function. The correction is to form likelihood times prior, then divide by its integral or sum over the parameter space. If only a proportional kernel is needed, state that it is proportional and do not interpret it as a normalized posterior.

The most useful check is to trace every factor to its role: the prior expresses population information for this patient before these measurements, the likelihood expresses the measurements conditional on individual parameters and design, and the evidence makes their product a probability distribution over those parameters. Keeping those roles separate makes the update interpretable and exposes assumptions that need clinical or model-based scrutiny.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

State the posterior and evidence for a continuous individual parameter vector, identifying which quantity is integrated over the parameter space.
A model uses three concentrations but assumes their errors are correlated conditional on the individual parameters. May its likelihood automatically be written as a product of three marginal densities? Explain.
For a two-point prior with weights 0.4 and 0.6 and likelihoods 0.5 and 0.1 for the observed data, calculate the evidence and both posterior probabilities.
Explain why the evidence does not depend on the unknown individual parameter, and why omitting it is not valid when reporting a normalized posterior.

Solutions. (1) The posterior is $p(\theta_i\mid\mathbf{y}_i,\mathcal{E}_i,x_i,\psi)=p(\mathbf{y}_i\mid\theta_i,\mathcal{E}_i)p(\theta_i\mid x_i,\psi)/m$, where $m=\int_{\Theta}p(\mathbf{y}_i\mid\theta_i,\mathcal{E}_i)p(\theta_i\mid x_i,\psi)\,d\theta_i$; the integral is over $\theta_i$. (2) No. Conditional dependence requires a joint likelihood that represents the correlation; multiplying marginal densities would impose conditional independence. (3) The unnormalized weights are $0.4(0.5)=0.20$ and $0.6(0.1)=0.06$, so the evidence is 0.26. The posterior probabilities are $0.20/0.26\approx0.769$ and $0.06/0.26\approx0.231$, which sum to one. (4) The evidence is obtained by integrating the likelihood-prior product over all parameter values, so after the observed data and conditioning variables are fixed it is a constant with respect to $\theta_i$. It is required to make the posterior integrate to one; without it the expression is only a proportional kernel.