Recursive Updates for Stable Individual Parameters

Why Recursion Can Preserve Individual Information

Therapeutic drug monitoring often adds concentration samples over time. If an individual’s pharmacokinetic (PK) parameter is assumed stable over the observations, new data should refine the same parameter estimate rather than replace it. Recomputing a posterior from every sample at once is conceptually simple, but a recursive update can produce the same result while retaining only a compact summary of the data.

The central issue is not whether the data arrive sequentially. It is whether the summary retained after each update contains all information needed for the next posterior. Under a conjugate model, this can be exact. For nonlinear PK likelihoods, compact summaries are often approximations, and applying an update to an already approximated posterior need not agree with fitting all observations together.

For example, let \(\theta\) be an individual’s log-clearance parameter, and let \(y_i\) be a concentration-derived measurement whose sampling model depends on \(\theta\). A common simplifying model is

This is a teaching model for the updating mechanics; a real concentration model may require a nonlinear prediction function, residual error model, and additional covariates. The normal model assumes the prior and conditional observation distributions are Gaussian, the observation variances \(R_i\) are known, and the observations are conditionally independent given the same \(\theta\). The sampling model and prior must remain the same as data are added.

Exact Gaussian Recursion

Suppose the posterior after \(k\) observations is \(N(m_k,V_k)\). For a new observation \(y_{k+1}\) with variance \(R_{k+1}\), multiplying the posterior density by the new likelihood gives another normal density. Its variance and mean are

The quantities \(1/V_k\) and \(m_k/V_k\) are the posterior precision and precision-weighted mean, respectively. Thus each independent observation adds its precision and its precision-weighted value. The update does not give each sample equal weight unless its measurement variance is the same as the others.

The equivalence to a batch calculation follows directly by induction. Initially, the prior contributes precision \(1/V_0\) and weighted mean \(m_0/V_0\). Each update adds \(1/R_i\) to precision and \(y_i/R_i\) to weighted mean. After \(n\) observations,

These are exactly the precision and weighted mean obtained by multiplying the prior density by all \(n\) likelihoods at once. The result depends on using the same observations, prior, and likelihood in either order of calculation. Because addition is commutative, the order in which conditionally independent samples are processed does not affect the final posterior.

What the Compact State Must Retain

For this Gaussian model, the pair consisting of posterior precision and precision-weighted mean is a sufficient compact state for continuing the recursion. Equivalently, one can retain \(m_k\) and \(V_k\), provided both are updated without rounding or other lossy transformation. Retaining only the posterior mean is not enough: two posteriors can have the same mean but different variances, and a new observation will affect them differently.

The same principle extends beyond the normal-normal example. In a conjugate exponential-family model, the posterior can often be represented by a fixed-dimensional set of natural parameters. When each new observation contributes additively to those parameters, retaining their updated values preserves the batch posterior. “Compact” therefore means a summary that is sufficient for the specified model, not an arbitrary short record of past data.

When Sequential Approximation Is Not Batch-Exact

Many individual PK models do not have a conjugate posterior. A concentration prediction may be nonlinear in clearance, volume, dose history, and time; the residual likelihood may also be nonnormal or heteroscedastic. In such cases, a convenient recursive method may approximate the current posterior by a normal distribution, a point estimate with curvature, or a small set of summary values. Updating that approximation is not generally the same operation as retaining the full posterior and multiplying by the next likelihood.

For example, a local Gaussian approximation based on a mode and curvature summarizes the shape near that mode. A later observation can move the mode into a region where the old approximation was poor. Updating only the old mode and curvature may then discard relevant asymmetry or multiple posterior modes. A batch fit using all observations can use the complete likelihood surface and need not equal the sequential approximation.

Compactness can still be useful if its purpose is explicit. One can retain the observations or likelihood contributions, sufficient statistics when available, or a carefully validated approximation with documented error. For an approximate update, compare sequential and batch calculations on representative simulated cases, including sparse data, extreme measurements, and weakly identified parameters. Agreement in one ordinary case is not proof of general equivalence.

Diagnostics, Alternatives, and Limitations

Before using a recursive update, verify that the model’s assumptions match the intended interpretation. For a stable individual parameter, observations should share the same latent parameter value over the update interval. Conditional independence should be defensible after conditioning on that parameter and modeled covariates. If residual errors remain correlated, multiplying separate marginal likelihoods as though they were independent can count information more than once and make the posterior too concentrated.

Useful checks include recomputing a small case in batch form, confirming that posterior variance decreases under the independent Gaussian model, and checking that a more precise observation has greater influence. A variance that increases in this model, or a recursive result that changes when the order of independent observations is reversed, signals an implementation or model inconsistency. Unexpectedly narrow uncertainty can indicate double-counting, ignored correlation, or excessive confidence in the measurement variance.

The main alternatives serve different purposes. Batch updating evaluates the prior times the full likelihood and is a useful reference when data are few or exact sequential summaries are unavailable. Exact conjugate recursion is computationally efficient and batch-equivalent under its assumptions. Approximate recursive schemes can reduce storage or computation for nonlinear models, but their agreement with batch inference must be assessed rather than presumed. If the parameter is allowed to evolve over time, the stable-parameter assumption no longer applies; that requires a model with an explicit evolution process rather than treating all samples as repeated measurements of one unchanged value.

A common mistake is to say, “The latest measurement is the best one, so replace the old posterior with it.” This discards prior information and previous likelihood contributions. The correction is to combine the current posterior with the new likelihood, using the new observation’s precision or the appropriate likelihood contribution. A different mistake is to assume that any compact summary is sufficient: exact preservation is a property of the model and the summary, not of the fact that an algorithm is recursive.