Sequential Bayesian Updating Across Concentration Samples

Why update one concentration at a time?

An individual’s concentration measurements often arrive at different times. A clinician may receive a trough result today and another sample after a dose adjustment later. Bayesian updating provides a direct way to incorporate each result: the posterior after one sample becomes the prior information for the next. The key requirement is that the update use the likelihood of the new observation conditional on what is already known, rather than treating old data as new evidence again.

The algebra can make sequential updating look different from fitting all available samples together. Under a correctly specified likelihood, however, the two calculations give the same posterior when they use the same prior and data. This equivalence is useful computationally, but it does not guarantee that the model is appropriate for a patient’s evolving pharmacokinetics.

Definition: Let $\theta$ denote an unknown individual parameter or parameter vector, $D_{1:i}$ the first $i$ concentration observations, and $x_i$ the known design information for observation $i$, such as dose and sampling time. Sequential Bayesian updating applies Bayes’ rule as each observation arrives: $$ p(\theta\mid D_{1:i},x_{1:i}) \propto p(y_i\mid \theta,D_{1:i-1},x_{1:i})\, p(\theta\mid D_{1:i-1},x_{1:i-1}). $$ Here $y_i$ is the new concentration observation. The likelihood factor must be conditional on the past data if the new observation is not conditionally independent of them.

From one observation to a sequence

For a typical concentration model, the observation distribution depends on the individual parameters and the sampling design. For example, on the log-concentration scale one might write $y_i=\log(C_i/(1\ \mathrm{mg/L}))$ and model $y_i$ around a prediction $f_i(\theta)$, which depends on dose, time, and the pharmacokinetic model. The prior describes uncertainty in $\theta$ before these individual observations; the likelihood describes how plausible the observed concentrations are for each value of $\theta$.

If observations are conditionally independent given $\theta$ and their known designs, their joint likelihood factors into individual contributions. Applying Bayes’ rule to that product gives the batch posterior. Applying Bayes’ rule repeatedly gives the same product, with one likelihood factor incorporated at each step.

Result: If the prior is the same in both calculations and the data likelihood factors as $p(D_{1:n}\mid\theta,x_{1:n})=\prod_{i=1}^n p(y_i\mid\theta,x_i)$, then $$ p(\theta\mid D_{1:n},x_{1:n}) \propto p(\theta)\prod_{i=1}^n p(y_i\mid\theta,x_i). $$ Sequential updating and batch updating therefore produce the same posterior, regardless of the order in which the factors are multiplied.

The factorization assumption is convenient, not automatic. If residuals are correlated, an observation’s likelihood generally depends on earlier observations conditional on the parameters. Sequential updating can still be exact, but it must use the appropriate conditional likelihood $p(y_i\mid\theta,D_{1:i-1},x_{1:i})$. Multiplying marginal likelihoods as if the residuals were independent would then give the wrong result.

Deriving the normal-normal update

A conjugate example makes the mechanics explicit. Suppose a scalar unknown $\theta$ represents a stable individual log-concentration level, and the measurements $y_i$ satisfy $y_i\mid\theta\sim N(\theta,s_i^2)$, with known measurement variances $s_i^2$. Assume the observations are conditionally independent given $\theta$, and the prior is $\theta\sim N(m_0,v_0)$. These assumptions are for this calculation: real pharmacokinetic predictions will usually depend on dose and time, and their likelihood need not reduce to this location model.

The prior density and each observation density contribute quadratic terms in $\theta$. Collecting those terms gives the posterior precision and mean below. Precision means inverse variance.

\begin{aligned} \frac{1}{v_n} &= \frac{1}{v_0}+\sum_{i=1}^n \frac{1}{s_i^2},\\ m_n &= v_n\left(\frac{m_0}{v_0}+ \sum_{i=1}^n \frac{y_i}{s_i^2}\right). \end{aligned}

To see why, the part of the joint density that depends on $\theta$ is proportional to $\exp\{-\tfrac12[(\theta-m_0)^2/v_0+\sum_i(y_i-\theta)^2/s_i^2]\}$. The coefficient of $\theta^2$ inside the brackets is $1/v_0+\sum_i1/s_i^2$, and the coefficient of the corresponding linear term is $-2[m_0/v_0+\sum_i y_i/s_i^2]$. Completing the square identifies the posterior variance as the inverse quadratic coefficient and the posterior mean as that variance times the weighted sum. Thus each new observation adds its precision and its precision-weighted value.

The same calculation can be applied recursively. If the current posterior is $N(m_{i-1},v_{i-1})$, then after observing $y_i$:

v_i=\left(\frac{1}{v_{i-1}}+\frac{1}{s_i^2}\right)^{-1}, \qquad m_i=v_i\left(\frac{m_{i-1}}{v_{i-1}}+\frac{y_i}{s_i^2}\right).

Worked Example: Updating a Log-Concentration Level

Suppose a hypothetical patient’s standardized log-concentration level has prior $\theta\sim N(1.2,0.25)$. Two measurements, already adjusted to represent this same level, are $y_1=1.0$ with measurement variance $s_1^2=0.16$, and $y_2=1.4$ with variance $s_2^2=0.04$. Assume the stated normal model and conditional independence. The log scale is defined as $y=\log(C/(1\ \mathrm{mg/L}))$, so it is dimensionless.

After the first observation, the prior precision is $1/0.25=4$, and the observation precision is $1/0.16=6.25$. Therefore $v_1=1/(4+6.25)=0.09756$, and $m_1=(1.2(4)+1.0(6.25))/10.25=11.05/10.25=1.078$.

For the second observation, its precision is $1/0.04=25$. The updated precision is $1/v_1+25=10.25+25=35.25$, so $v_2=1/35.25=0.02837$. The updated mean is $m_2=(1.07805(10.25)+1.4(25))/35.25=(11.05+35)/35.25=1.306$. The posterior standard deviation is $\sqrt{0.02837}=0.1684$. The second, more precise observation pulls the mean toward 1.4 and reduces posterior uncertainty. These are posterior summaries for the hypothetical level under the stated model, not a clinical dose recommendation.

Worked Example: Verifying Batch and Sequential Equivalence

Use a hypothetical prior $\theta\sim N(0,1)$ and two independent measurements: $y_1=0.5$ with variance $0.25$, and $y_2=1.0$ with variance $1$. These may represent standardized log-concentration observations under the simple location model above.

In a batch calculation, the total precision is $1+1/0.25+1/1=6$, so the posterior variance is $1/6=0.1667$. The weighted sum is $0(1)+0.5(4)+1.0(1)=3$, giving posterior mean $3/6=0.5000$.

Sequentially, the first update gives variance $1/(1+4)=0.2$ and mean $(0(1)+0.5(4))/5=0.4$. The second update gives variance $1/(5+1)=0.1667$ and mean $(0.4(5)+1.0(1))/6=3/6=0.5000$. Both routes yield $N(0.5000,0.1667)$. The agreement follows from using the same prior and both likelihood factors exactly once.

Sequential computation with a nonlinear PK likelihood

For a nonlinear pharmacokinetic model, a closed-form normal update may not exist. The defining update is still Bayes’ rule: multiply the current posterior density by the new observation’s conditional likelihood and normalize. In numerical work, this can be implemented using a grid, quadrature, or posterior samples. The next example uses a small grid to make the multiplication visible; the two candidate parameter sets stand in for points at which a continuous model’s likelihood might be evaluated.

Worked Example: Updating Two Candidate PK Parameter Sets

Consider two hypothetical candidate parameter sets, A and B, with prior probabilities 0.60 and 0.40. A first concentration has relative likelihood contributions 0.80 and 0.30 under A and B. A second concentration has conditional relative likelihood contributions 0.25 and 0.75, respectively. Assume these are the correct likelihood factors for the observations given each candidate and that omitted constants are common across candidates.

After the first sample, the unnormalized weights are $0.60(0.80)=0.48$ for A and $0.40(0.30)=0.12$ for B. Their sum is 0.60, so the normalized posterior weights are $0.48/0.60=0.80$ and $0.12/0.60=0.20$.

After the second sample, multiply those weights by its likelihood contributions: A has $0.80(0.25)=0.20$, and B has $0.20(0.75)=0.15$. Normalizing by $0.35$ gives $P(A\mid D_{1:2})=0.20/0.35=0.5714$ and $P(B\mid D_{1:2})=0.15/0.35=0.4286$.

The batch calculation gives the same result: the unnormalized weights are $0.60(0.80)(0.25)=0.12$ and $0.40(0.30)(0.75)=0.09$; dividing by $0.21$ again gives 0.5714 and 0.4286. The second sample favors B, but the final posterior still assigns slightly more weight to A because the prior and first sample favored A.

Conditions, diagnostics, and alternatives

In practice, exact sequential-batch equivalence requires consistent treatment of the prior, data, and likelihood. The sampling times, doses, and any relevant dose changes must be represented in the concentration predictions. If a sample is selected adaptively after earlier results, the design should be handled appropriately: when the selection rule depends only on recorded past information and adds no unmodeled information about $\theta$, the concentration likelihood can be updated conditional on that design. If the sampling process itself carries information about the parameter and is not ignorable, it may need to be modeled too.

Useful checks include confirming that each observation appears once, verifying that the new likelihood is evaluated at the actual dose and sampling time, and checking that posterior weights or densities normalize. In the conjugate model, each independent observation with finite variance adds positive precision, so posterior variance must decrease. A variance that increases under that model signals an arithmetic or implementation error. In a general PK model, uncertainty in a particular parameter need not decrease monotonically: parameter trade-offs, nonlinear likelihood shapes, or changing model structure can alter marginal uncertainty.

A practical comparison is to compute the posterior from all observations at once on a small test case and compare it with the sequential result. Disagreement can reveal duplicated data, inconsistent priors, a missing conditional dependence, or different numerical approximations. Agreement is a calculation check, not proof that the pharmacokinetic model fits the patient.

Approach	Strength	Important qualification
Sequential updating	Incorporates results as they arrive and can carry the current posterior forward.	Requires the correct conditional likelihood and careful avoidance of double counting.
Batch updating	Uses all observations in one calculation and makes the joint likelihood explicit.	May not be available at the time an early clinical decision is needed.
Approximate numerical updating	Can handle nonlinear models where analytic formulas are unavailable.	Grid resolution, numerical approximation, and posterior coverage require checking.

Measurements from repeated occasions, correlated individual effects, and time-varying parameters may require a richer joint model than a single stable $\theta$. Earlier material on correlated individual random effects and time-varying PK parameters gives relevant context. Simply applying an independent, stable-parameter update to data generated under dependence or parameter change can produce misleading certainty or shift the posterior for the wrong reason.

Common mistake: treating the current posterior as new prior information

A common error is to carry forward a posterior based on the first sample and then update with both the first and second samples again. That counts the first likelihood factor twice. The correct sequential step multiplies the current posterior by the likelihood contribution from the new observation only. Another error is to assume that sequential and batch calculations must agree even when the sequential method uses independent likelihood factors but the observations are correlated. Equivalence holds for the same joint likelihood, not for a convenient but incorrect factorization.

Key takeaway: A new concentration updates the current posterior through its appropriate conditional likelihood. When the same prior and correctly specified joint likelihood are used, sequential and batch updating are algebraically equivalent; the order of calculation does not repair a misspecified model.

Check Your Understanding

Work through each question, then compare your reasoning with the solutions.

Under the normal-normal model, a prior has mean 2 and variance 1. A new observation is 3 with variance 1. Find the posterior mean and variance.
Why can multiplying separate marginal likelihoods give an incorrect sequential update when residuals are correlated?
A sample was used to obtain the current posterior. What likelihood contribution should be used when the next sample arrives, and what happens if the old sample is included again?
State the conditions that make sequential and batch updating equivalent, and explain why equivalence does not establish that the PK model is correct.

Solutions. (1) Prior and observation precisions are both 1, so the posterior variance is $1/(1+1)=0.5$, and the mean is $(2(1)+3(1))/2=2.5$. (2) With correlation, the conditional distribution of the new observation generally depends on the past observations given the parameters. Separate marginal likelihoods discard that dependence and do not equal the joint likelihood. (3) Multiply the current posterior by the likelihood of the new observation conditional on the past and the design. Including the old sample again counts its evidence twice and generally makes the posterior too concentrated or shifts it incorrectly. (4) Use the same prior and the same correctly specified joint likelihood, factoring it into valid conditional contributions; then multiplication order does not change the result. Both calculations can agree while sharing a misspecified concentration model, so agreement checks computation rather than model adequacy.