Sequential Bayesian Updating After New Concentrations

Why Updating Must Preserve the Existing Posterior

Therapeutic drug monitoring often produces information in stages: an initial set of concentrations informs a patient-specific parameter distribution, and a later concentration becomes available after another dose or sampling occasion. The new result should refine the existing distribution, not replace it with an estimate based only on the latest observation. The key is to combine the current posterior with the likelihood of the new data, while preserving any dependence created by shared patient or occasion effects.

Let \(\theta\) denote the patient-specific model parameters, such as log clearance and log volume, and let \(y_{1:k}=(y_1,\ldots,y_k)\) denote the concentrations already observed. The current posterior \(p(\theta\mid y_{1:k})\) summarizes the parameter uncertainty after those observations, under the model and prior used so far. When a new observation \(y_{k+1}\) arrives, the general update is

p(\theta\mid y_{1:k+1}) \propto p(y_{k+1}\mid \theta,y_{1:k})p(\theta\mid y_{1:k}).

The proportionality constant is chosen so that the updated distribution integrates to one. The likelihood term is conditional on the earlier observations as well as on \(\theta\). It can be simplified to \(p(y_{k+1}\mid\theta)\) only when the observations are conditionally independent given \(\theta\) alone. If they share a latent occasion effect, uncertain dose history, or another unobserved quantity, that quantity may need to remain in the conditioning or be included in a joint update.

Definition: A sequential Bayesian update uses the posterior after the data currently available as the prior distribution for the next update, then multiplies it by the new observation's correct conditional likelihood. The update carries forward a distribution, not merely a point estimate.

The Update Rule and Its Justification

Bayes' rule for the full data is \(p(\theta\mid y_{1:k+1})\propto p(y_{1:k+1}\mid\theta)p(\theta)\). The chain rule for probabilities gives

p(y_{1:k+1}\mid\theta) = p(y_{k+1}\mid\theta,y_{1:k})p(y_{1:k}\mid\theta).

Substituting this factorization into the full-data posterior gives

p(\theta\mid y_{1:k+1}) \propto p(y_{k+1}\mid\theta,y_{1:k}) p(y_{1:k}\mid\theta)p(\theta).

But \(p(y_{1:k}\mid\theta)p(\theta)\), after normalization, is exactly \(p(\theta\mid y_{1:k})\). This yields the sequential rule. If \(y_1,\ldots,y_{k+1}\) are conditionally independent given \(\theta\) alone, the conditional likelihood simplifies to \(p(y_{k+1}\mid\theta)\). In that case, multiplication of the likelihood contributions is associative: processing observations one at a time gives the same posterior as analyzing all observations together, provided the same prior and model are used.

This equivalence does not mean every compressed summary is sufficient for updating. The full posterior distribution is a valid carry-forward object. A mean and variance are sufficient only in special model families where those quantities determine the posterior and the update can be performed exactly from them. For a skewed or multimodal posterior, retaining only its mean and variance can lose information that matters when the next likelihood is applied.

Worked Example: Two Independent Concentration Updates

Suppose a hypothetical patient has a dimensionless log-clearance deviation \(\theta=\log(CL/CL_{\mathrm{ref}})\). After adjusting each concentration for the known dose and sampling-time prediction, define \(z\) as the log concentration ratio to that prediction. Use the model \(z=-\theta+\epsilon\), where \(\epsilon\sim N(0,0.01)\). Assume the errors for the two samples are independent given \(\theta\), and start with \(\theta\sim N(0,0.04)\). The first adjusted result is \(z_1=-0.20\), and the later result is \(z_2=0.10\).

For an observation \(z=-\theta+\epsilon\), the equivalent noisy measurement of \(\theta\) is \(-z=\theta-\epsilon\), with measurement variance \(0.01\). For normal prior variance \(V\) and measurement variance \(R\), the updated precision is \(1/V+1/R\), and the updated mean is the precision-weighted average of the prior mean and the implied measurement \(-z\).

For the first result, the implied measurement is \(-z_1=0.20\). The prior precision is \(1/0.04=25\), and the observation precision is \(1/0.01=100\). Therefore,

V_1=\frac{1}{25+100}=0.008,\qquad \mu_1=\frac{25(0)+100(0.20)}{125}=0.16.

Thus the posterior after the first concentration is \(N(0.16,0.008)\). For the second result, the implied measurement is \(-z_2=-0.10\). Updating that posterior gives

V_2=\frac{1}{125+100}=\frac{1}{225}=0.004444, \qquad \mu_2=\frac{125(0.16)+100(-0.10)}{225} =\frac{10}{225}=0.04444.

The final posterior standard deviation is \(\sqrt{1/225}=0.06667\). A 95% normal posterior interval is approximately \(0.04444\pm1.96(0.06667)=[-0.08622,0.1751]\). Under this model, the interval describes posterior uncertainty about the patient's log-clearance deviation; it is not a 95% probability statement about a fixed parameter under repeated sampling. The two results pull in opposite directions, so the final mean is between their implied values. This update is valid because the assumed observation errors are independent given \(\theta\).

Shared Effects Change the Conditional Likelihood

When two concentrations share an unobserved effect, they may be dependent even after conditioning on the patient parameter. In that case, using a marginal likelihood for the new concentration as if it were independent of the earlier one can count the shared information twice. A correct sequential update can retain the latent effect in the joint posterior, or use the conditional likelihood after accounting for the earlier observation.

Worked Example: Updating With a Shared Occasion Effect

Consider two hypothetical standardized log-concentration residuals \(y_1=y_2=2\), measured on the same occasion. Let \(\theta\) be a dimensionless patient-level offset with prior \(\theta\sim N(0,1)\). Suppose \(y_j=\theta+b+\epsilon_j\), where the shared occasion effect \(b\sim N(0,1)\) and independent residuals \(\epsilon_j\sim N(0,1)\) are mutually independent and independent of \(\theta\). The shared \(b\) makes the two observations dependent given \(\theta\).

For the first observation, integrating over \(b\) and \(\epsilon_1\) gives \(y_1\mid\theta\sim N(\theta,2)\). Combining this likelihood with the prior gives posterior precision \(1+1/2=3/2\), variance \(2/3\), and mean

E(\theta\mid y_1)= \frac{1(0)+(1/2)(2)}{1+1/2} =\frac{2}{3}.

For the second update, the relevant likelihood is \(p(y_2\mid\theta,y_1)\), not the marginal \(p(y_2\mid\theta)\). Given \(\theta\) and \(y_1=2\), the posterior distribution of \(b\) is normal with mean \((2-\theta)/2\) and variance \(1/2\): this follows by combining the \(N(0,1)\) prior on \(b\) with the \(N(\theta+b,1)\) likelihood for \(y_1\). Consequently,

y_2\mid\theta,y_1=2 \sim N\left(\frac{\theta+2}{2},\frac{3}{2}\right).

The second result can therefore be viewed as the noisy measurement \(y_2-1=1\) of \(\theta/2\), with variance \(3/2\), or equivalently as a measurement of \(\theta\) with value \(2\) and variance \(6\). Its precision for \(\theta\) is \(1/6\). Updating the first posterior gives

V_2=\frac{1}{3/2+1/6}=0.6,\qquad \mu_2=\frac{(3/2)(2/3)+(1/6)(2)}{3/2+1/6}=0.8.

Thus \(\theta\mid y_1,y_2\sim N(0.8,0.6)\). If the second observation were incorrectly treated as an independent \(N(\theta,2)\) observation, it would add precision \(1/2\), rather than the correct conditional precision \(1/6\), and produce an overconfident posterior. The shared occasion effect means that observing the first residual already provides information about the effect contributing to the second.

Updating Several Patient Parameters

The same principle applies when \(\theta\) is a vector, for example a patient's log clearance and log volume. In a multivariate normal model, updating means adding the new observation's information to the current posterior precision and updating the corresponding precision-weighted mean. The geometry matters: a concentration may inform a combination of parameters more strongly than either parameter separately.

Worked Example: Two Concentrations Inform Different Parameter Combinations

Let the hypothetical patient parameter vector be \(\theta=(\theta_1,\theta_2)^\mathsf{T}\), with prior \(N(\mathbf{0},I)\), where \(I\) is the \(2\times2\) identity matrix. Suppose two independent, standardized, model-adjusted concentration measurements follow \(y_1=\theta_1+\theta_2+\epsilon_1\) and \(y_2=\theta_1-\theta_2+\epsilon_2\), with \(\epsilon_1,\epsilon_2\sim N(0,0.25)\). Let the observed values be \(y_1=1.2\) and \(y_2=-0.2\).

For the first observation, the design vector is \(h_1=(1,1)^\mathsf{T}\). The prior precision is \(I\); the observation adds \(h_1h_1^\mathsf{T}/0.25\). Thus the first posterior precision and mean are

\Lambda_1=I+4 \begin{pmatrix}1&1\\1&1\end{pmatrix} = \begin{pmatrix}5&4\\4&5\end{pmatrix}, \qquad \mu_1=\Lambda_1^{-1}(4y_1h_1) = \begin{pmatrix}0.5333\\0.5333\end{pmatrix}.

The second observation has design vector \(h_2=(1,-1)^\mathsf{T}\), so it adds \(4h_2h_2^\mathsf{T}\) to the precision and \(4y_2h_2\) to the precision-weighted mean. Therefore,

\Lambda_2= \begin{pmatrix}5&4\\4&5\end{pmatrix} + 4\begin{pmatrix}1&-1\\-1&1\end{pmatrix} = \begin{pmatrix}9&0\\0&9\end{pmatrix}, $$ $$ \mu_2=\Lambda_2^{-1} \left[ \begin{pmatrix}5&4\\4&5\end{pmatrix} \begin{pmatrix}0.5333\\0.5333\end{pmatrix} +4(-0.2)\begin{pmatrix}1\\-1\end{pmatrix} \right] = \begin{pmatrix}4/9\\5.6/9\end{pmatrix} = \begin{pmatrix}0.4444\\0.6222\end{pmatrix}.

The updated covariance is \(\Lambda_2^{-1}=\operatorname{diag}(1/9,1/9)\). The first result primarily informed the sum of the parameters; the second informed their difference. In this example the two combinations together yield independent parameter uncertainty in the final posterior. In an actual pharmacokinetic model, the design vectors and likelihood may be nonlinear, but the same requirement holds: the update must use the likelihood induced by the model for all parameters being inferred.

Practical Checks, Alternatives, and Limitations

A reliable update starts by checking what the current posterior represents and what information the new concentration adds. Confirm that the patient-specific prior or posterior uses the same parameter definitions, dose record, covariates, and likelihood assumptions as the new update. The sampling time and dose history determine the concentration likelihood; an apparently surprising concentration can reflect a misspecified input history as well as a changed parameter estimate. The relevant conditional independence assumption must also be justified rather than inferred merely because the observations were collected at different times.

For a conjugate model, the posterior parameters can often be updated analytically, as in the examples. For a nonlinear pharmacokinetic likelihood, one may instead evaluate the new likelihood at draws from the existing posterior and reweight those draws. This is appropriate only if the draws represent the full current posterior and the new likelihood is correctly conditional on the earlier data. If the new likelihood is concentrated where the existing posterior has almost no draws, simple reweighting may be unreliable; refitting the full model or using an appropriate sampling method may be needed.

A batch refit and a sequential update are equivalent when they use the same prior, model, and complete data likelihood, and when the sequential likelihood is correctly conditional. A batch refit is often easier when the model changes or shared latent variables must be reconstructed. A sequential update is convenient when data arrive over time, but it does not justify discarding past uncertainty. A plug-in update that substitutes only a posterior mean for \(\theta\) ignores parameter uncertainty and generally does not equal Bayesian updating.

Warning signs include a posterior that becomes much narrower after a highly correlated new sample, a result that changes substantially when the order of independent observations is reversed, or a sequential result that disagrees with a batch calculation under the same model. Order dependence can be legitimate if the conditional likelihoods differ because of dependence or changing information; it is not expected when the same conditionally independent likelihood contributions are multiplied correctly. These checks diagnose possible errors but do not establish that the pharmacokinetic model itself is adequate.

Finally, the posterior is conditional on its assumptions. Incorrect residual-error structure, unmodeled occasion effects, inaccurate sampling times, or a poor population prior can yield a precise but misleading update. Sequential updating carries forward the consequences of earlier modeling choices; it does not automatically repair them.

Mistake: Treating the latest concentration as an independent likelihood contribution simply because it was measured later. Correction: Check whether observations are independent given the patient parameters alone; if not, condition on or jointly update the relevant shared latent quantities.
Mistake: Carrying forward only a posterior mean and treating it as known. Correction: Preserve the posterior distribution, or use a summary only when the model provides an exact update from that summary.
Mistake: Concluding that a sequential posterior must be identical to a batch posterior under any workflow. Correction: The equivalence requires the same model and prior, and a correct factorization of the full likelihood.

Key takeaway: Update the existing patient posterior with the new observation's correct conditional likelihood. The simple product with \(p(y_{k+1}\mid\theta)\) is valid only when the new observation is conditionally independent of earlier data given \(\theta\) alone.

Check Your Understanding

Answer each question, then compare with the solutions.

A new concentration is conditionally independent of all earlier concentrations given \(\theta\). What likelihood is used in the sequential update?
In the shared-effect example, why is the second likelihood not simply \(p(y_2\mid\theta)\)?
A normal prior for a scalar parameter has mean \(0\) and variance \(1\). An independent measurement of that parameter is \(1\) with error variance \(1\). Find the updated mean and variance.
When should a batch and sequential analysis give the same posterior?

Solutions. (1) Use \(p(y_{k+1}\mid\theta)\), because conditional independence given \(\theta\) alone makes the earlier observations irrelevant to the new likelihood once \(\theta\) is fixed. (2) The observations share the latent occasion effect \(b\); the first result informs \(b\), so the correct second likelihood is conditional on both \(\theta\) and \(y_1\), or the joint posterior must retain \(b\). (3) The prior precision and observation precision are both \(1\), so the posterior variance is \(1/(1+1)=0.5\), and the mean is \([1(0)+1(1)]/2=0.5\). (4) They agree when both use the same prior, model, and data likelihood, with each sequential step using the correct conditional likelihood.