Sequential Sampling Design With Updated Information

Why Update Information Sequentially?

A fixed sampling schedule must be specified before observations begin. In a sequential design, instead, the next sampling time can depend on information already collected: a concentration measured early in a profile may help determine when a later sample is most useful. The design question is therefore conditional: given the history observed so far, which feasible next time is expected to contribute most to the chosen objective?

The key distinction is between information conditional on the history that actually occurred and unconditional information averaged over all histories that the policy could produce. These are not interchangeable when the selected sampling time varies with history. This tutorial makes that distinction explicit and uses it to construct one-step sequential rules.

Conditional and Unconditional Information

Let $\theta$ be the vector of NLME parameters of interest, and let $H_k$ denote the history after $k$ observations, including observed data and the sampling times already used. A sequential rule chooses the next time as $t_{k+1}=d(H_k)$, subject to the allowed sampling constraints. The rule is assumed fixed in advance and to depend on observed history, not directly on the unknown value of $\theta$. Any randomization in the rule must likewise have a known, parameter-independent distribution.

Write $Y_{k+1}$ for the next observation and $p(y\mid H_k,t;\theta)$ for its conditional likelihood under the NLME model. This is the appropriate marginal likelihood if random effects are latent and integrated out. Define the conditional score and conditional Fisher information for a proposed time $t$ by

u_{k+1}(y\mid H_k,t;\theta) = \frac{\partial}{\partial\theta}\log p(y\mid H_k,t;\theta), \qquad J(t\mid H_k;\theta) = E\!\left[ u_{k+1}u_{k+1}^{T}\mid H_k,t \right].

Under regularity conditions, the conditional score has mean zero given $H_k$ and $t$. The conditions include a differentiable likelihood, a support that does not depend on $\theta$, and sufficient integrability to interchange differentiation and integration. Thus $J(t\mid H_k;\theta)$ is the expected information in the next observation conditional on the history and proposed time; it is not the information from the particular value of $Y_{k+1}$ that will be observed.

Definition: Let $S_k$ be the score of the full likelihood for the data collected through step $k$, and define the unconditional Fisher information $I_k=E[S_kS_k^T]$, with expectation over all histories generated by the design policy. If $t_{k+1}=d(H_k)$, the update is $$ I_{k+1}=I_k+E_{H_k}\!\left[J\!\left(d(H_k)\mid H_k;\theta\right)\right]. $$ The expectation includes any randomization in the sampling rule. For a realized history, $J(d(H_k)\mid H_k;\theta)$ is a conditional increment; by itself it is not the unconditional information increment.

Why the Information Update Holds

The update follows from the factorization of the sequential likelihood into the likelihood through step $k$ and the conditional likelihood of the next observation. Consequently, the full score is $S_{k+1}=S_k+u_{k+1}$. Because $S_k$ is determined by the history, it is fixed when conditioning on $H_k$. The zero-mean conditional score then makes the cross-products vanish in expectation.

\begin{aligned} E[S_{k+1}S_{k+1}^{T}] &=E[S_kS_k^{T}] +E[u_{k+1}u_{k+1}^{T}]\\ &\quad +E[S_ku_{k+1}^{T}] +E[u_{k+1}S_k^{T}],\\ E[S_ku_{k+1}^{T}] &=E\!\left[S_kE(u_{k+1}^{T}\mid H_k)\right]=0,\\ E[u_{k+1}S_k^{T}] &=E\!\left[E(u_{k+1}\mid H_k)S_k^{T}\right]=0,\\ E[u_{k+1}u_{k+1}^{T}] &=E_{H_k}\!\left[J(d(H_k)\mid H_k;\theta)\right]. \end{aligned}

Combining these identities gives the stated update. It is exact for unconditional Fisher information under the assumptions above. If the next time differs across histories, its conditional information also differs; the unconditional increment is the average of those conditional matrices, not the matrix for one selected history.

Worked Example: Averaging History-Dependent Increments

Suppose the information accumulated through step $k$ is $I_k=12$ in a one-parameter design. Two types of history occur: with probability $0.25$, the policy selects a later time whose conditional information is $6$; with probability $0.75$, it selects an earlier time whose conditional information is $2$. These probabilities and information values are hypothetical.

The expected increment is \[ E[J]=0.25(6)+0.75(2)=1.5+1.5=3. \] Thus the unconditional information after the next observation is \[ I_{k+1}=12+3=15. \] If $I_k(h)$ denotes information through step $k$ conditional on history $h$, the history-specific totals are $I_k(h_{\text{later}})+6$ and $I_k(h_{\text{earlier}})+2$, respectively; their values cannot be determined from the unconditional $I_k=12$ alone. In particular, writing $I_{k+1}=12+6$ would incorrectly apply the increment from only one possible history to the whole policy.

Selecting a Next Time by Expected Criterion Gain

A sampling criterion maps an information matrix to a measure of design quality. Common choices include D-optimality, which uses $\log\det(M)$ for a positive-definite information matrix $M$; A-optimality, which seeks to reduce $\operatorname{tr}(M^{-1})$; and E-optimality, which seeks to increase the smallest eigenvalue. These criteria emphasize different aspects of parameter precision, so the intended objective must be selected before choosing a time.

For a one-step rule at an observed history $H_k=h$, let $M_h$ be the information matrix being used as the current design state, and let $J(t\mid h;\theta)$ be the expected information from a candidate time. A common Fisher-information rule selects the feasible $t$ that maximizes the criterion gain \[ \Delta_{\Phi}(t\mid h)=\Phi\!\left(M_h+J(t\mid h;\theta)\right)-\Phi(M_h). \] This is a gain in the chosen criterion evaluated on expected information. It is not generally the exact expected gain in a nonlinear function of the information from the realized next observation. If the objective is instead the expected value of a realized-data utility, that utility must be defined and averaged over the possible next observations.

For a policy evaluated across possible histories, the unconditional Fisher matrix uses $I_k+E[J(d(H_k)\mid H_k;\theta)]$. In general, evaluating a nonlinear criterion at that average matrix is not the same as averaging conditional criterion gains. That difference matters whenever a design policy produces substantially different information across histories.

Worked Example: Choosing Between Two Times by D-Optimality

At a particular history, suppose the current two-parameter design information is $M_h=\operatorname{diag}(10,4)$. Two feasible candidate times have conditional expected information matrices $J_A=\operatorname{diag}(4,0.2)$ and $J_B=\operatorname{diag}(0.5,3)$. Treat the parameters as standardized, so the entries are on a common dimensionless scale, and use the D-optimal criterion $\Phi(M)=\log\det(M)$.

For time A, the updated matrix is $\operatorname{diag}(14,4.2)$, with determinant $14(4.2)=58.8$. Its criterion gain is \[ \Delta_D(A\mid h)=\log(58.8)-\log(40)=\log(1.47)\approx0.3853. \] For time B, the updated matrix is $\operatorname{diag}(10.5,7)$, with determinant $10.5(7)=73.5$. Its gain is \[ \Delta_D(B\mid h)=\log(73.5)-\log(40)=\log(1.8375)\approx0.6084. \] Both matrices are positive definite, so the log determinants are defined. Under this one-step D-optimal rule, choose time B because its expected information produces the larger log-determinant gain. The result is specific to this criterion and history; it does not establish that B is best under every objective or for every possible history.

Worked Example: Criterion of Average Information Versus Average Gain

Suppose the current scalar information is $M=4$. Across two equally likely histories, an adaptive policy produces a next-observation information increment of $J=4$ in one history and $J=0$ in the other. Use the scalar D-optimal criterion $\Phi(M)=\log M$.

The unconditional information increment is $E[J]=0.5(4)+0.5(0)=2$, so the information averaged across histories is $4+2=6$. The criterion gain evaluated at this average information is \[ \log(6)-\log(4)=\log(1.5)\approx0.4055. \] The average of the history-specific criterion gains is instead \[ 0.5\{\log(8)-\log(4)\}+0.5\{\log(4)-\log(4)\} =0.5\log(2)\approx0.3466. \] The values differ because the logarithm is nonlinear. The first quantity is the gain from applying the criterion to the unconditional average information; the second is the average of conditional gains. A design analysis must state which quantity is its objective rather than treating them as equal.

Assumptions, Diagnostics, and Failure Modes

A sequential information calculation is only as reliable as the conditional likelihood and the design rule used to obtain it. Before using the update, check the following:

Correct conditional model. The distribution $p(y\mid H_k,t;\theta)$ must reflect the dependence induced by the NLME model, including any relevant random effects and residual-error structure. Information from a conditional likelihood that is not the likelihood used for inference can be misleading.
Valid score regularity. Check differentiability, parameter-independent support, finite score second moments, and a zero conditional expected score. Boundary parameters or nonregular likelihoods can invalidate the usual information identity.
Feasible and defined candidates. Candidate times must satisfy operational constraints, and the resulting information matrices must be suitable for the criterion. For example, $\log\det(M)$ and $M^{-1}$ require positive definiteness.
History dependence is retained. When simulating or enumerating adaptive policies, calculate the conditional increment for each history and average over histories for the unconditional matrix. Replacing this with a single “typical” history can distort both the update and the ranking of times.
Numerical stability is checked. Inspect eigenvalues and symmetry of computed information matrices. Very small eigenvalues can indicate weakly informed parameter directions or numerical problems; a nominally large determinant alone does not diagnose every weak direction.

Monte Carlo evaluation is useful when the distribution of histories is too complicated to enumerate. Simulate histories under the current policy, compute $J(d(H_k)\mid H_k;\theta)$ for each, and average the matrices. Check that increasing the number of simulated histories stabilizes the entries and the resulting design decision. Such checks assess numerical stability, not whether the model or policy assumptions are correct.

How Sequential Design Differs From Alternatives

Approach	How sampling times are chosen	Main distinction
Fixed design	All times are set before data collection.	Does not adapt to observed histories; information can be calculated for the prespecified schedule.
One-step sequential design	At each step, choose the feasible time with the largest conditional expected criterion gain.	Uses current history but may not account fully for later opportunities.
Multi-step look-ahead design	Evaluate a candidate time by accounting for possible future observations and later decisions.	Can reflect downstream value, but requires more computation and a model for future histories and decisions.

The one-step rule is attractive when decisions must be made promptly or a full look-ahead calculation is impractical. Its limitation is that the most informative immediate observation need not lead to the best final schedule. It can also be sensitive to the nominal parameter values used to calculate $J$. If those values are uncertain, information gains may be evaluated over plausible parameter values or scenarios; the resulting rule should specify how those scenarios are weighted.

A further limitation is that Fisher information is an expected, local measure. It does not guarantee accurate estimation in a finite sample, protect against model misspecification, or ensure successful estimation when parameters are weakly identified. Sequential adaptation can also make actual sampling times depend on data, so the likelihood and reported design must reflect the rule that was followed.

Common Pitfall: Treating a Conditional Increment as Unconditional

A tempting argument is: “The cross-products between the old score and the next score are zero, so the total information is the old information plus $J(t\mid H_k;\theta)$.” The cross-products do vanish under the stated conditions, but if $t$ and $J$ vary with the random history, the latter expression is still history-dependent. It is not the unconditional information matrix.

Correction: For unconditional Fisher information under an adaptive policy, average the conditional increment over the histories and any randomization: $$ I_{k+1}=I_k+E_{H_k}\!\left[J(d(H_k)\mid H_k;\theta)\right]. $$ For a realized history, $J(d(H_k)\mid H_k;\theta)$ is the conditional expected contribution at that history. Use it for a conditional decision; do not report it as the unconditional increment without averaging.

Check Your Understanding

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

A policy yields conditional information increments of $2$ and $8$ with probabilities $0.75$ and $0.25$. What is the unconditional increment?
Why do cross-products between the accumulated score and the next conditional score have expectation zero?
For scalar current information $M=5$, compare the log-information gains from two candidate increments, $J_A=1$ and $J_B=3$. Which does the one-step D-optimal rule select?
State why the average of conditional log-determinant gains need not equal the log-determinant gain calculated from average information.
Name two conditions that should be checked before relying on the conditional Fisher-information update.

Solutions. (1) $0.75(2)+0.25(8)=1.5+2=3.5$. (2) The accumulated score is fixed given the history, while the next conditional score has conditional mean zero; iterated expectation therefore gives a zero cross-product. (3) The gains are $\log(6/5)=\log(1.2)\approx0.1823$ and $\log(8/5)=\log(1.6)\approx0.4700$, so the rule selects B. (4) The log determinant is nonlinear, so applying it after averaging matrices generally differs from averaging its values across histories. (5) Examples include differentiability and parameter-independent support of the likelihood, finite score second moments, a zero conditional expected score, and a valid parameter-independent sampling rule conditional on observed history.