From a Filtered State to the Next Observed Concentration
At a sampling time, the concentration recorded by an assay is not generally the SDE state itself. The state is uncertain because it was only indirectly observed, it evolves stochastically between samples, and the assay may add further error. A one-step predictive distribution combines these sources in the order the data-generating process does: begin with the filtered state at the last observation, propagate it across the next sampling interval, and then apply the observation model.
This distribution answers a specific question: given the observations available through time \(t_{j-1}\), what distribution does the model assign to the observation \(Y_j\) at time \(t_j\)? It is a forecast made before using \(Y_j\) to update the state. It is therefore different from the filtered distribution after \(Y_j\) has been incorporated.
Write the filtered distribution as \(\pi_{j-1}(dx)\). Let \(K_j(x,dx')\) be the SDE transition distribution from state \(x\) at \(t_{j-1}\) to state \(x'\) at \(t_j\), conditional on the known dose history and other inputs in that interval. Let \(g_j(y\mid x')\) be the observation density at \(t_j\), conditional on the state \(x'\). The usual assumptions are that the SDE and observation model define these conditional distributions, the inputs are known or conditioned upon, and the observation at \(t_j\) is conditionally independent of earlier observations given the current state and model parameters. If parameters are held fixed, these assumptions give
This equation is the predictive calculation in its general form. The inner integration propagates the filtered state through the SDE; the outer integration accounts for uncertainty in the state at the previous observation time. The observation density is then averaged over the resulting predicted state distribution. If densities exist, the predicted state density is
Known doses or other inputs may change the transition distribution within the interval. The transition must therefore reflect the actual event times and input history; it is not enough to propagate across the elapsed time using a transition that assumes no intervening dose.
Two Useful Derivations
The predictive density follows by conditioning on the state at the observation time. For any measurable set \(A\), the law of total probability gives
By the conditional-independence assumption, the first conditional probability is determined by the observation model, and the second distribution is obtained by propagating the filtered state through the SDE. If the observation model has density \(g_j\), this identity yields the predictive-density formula above. The argument uses conditional probability and integration; it does not require the predictive distribution to be Gaussian.
For the mean, condition first on \(X_{t_j}\):
For the variance, subtract the predictive mean and add and subtract the conditional mean:
The conditional expectation of the first term given \(X_{t_j}\) and \(\mathcal H_{j-1}\) is zero. Consequently, the cross-product has expectation zero, and the expected square of the first term is \(E[v(X_{t_j})\mid\mathcal H_{j-1}]\). The expected square of the second term is \(\operatorname{Var}(m(X_{t_j})\mid\mathcal H_{j-1})\). Adding these terms proves the variance identity. The first variance contribution is conditional observation variability; the second is variability in the observation mean induced by uncertainty in the predicted state. \(\square\)
Worked Examples
Worked Example: Linear SDE With Gaussian Assay Error
Suppose the state follows the Ornstein–Uhlenbeck SDE \(dX_t=-0.5X_t\,dt+0.4\,dW_t\) over a one-unit sampling interval, with no dose in that interval. At the previous sampling time, \(X\mid\mathcal H_{j-1}\sim N(1,0.25)\). The observation model is \(Y_j=X_{t_j}+\epsilon_j\), where \(\epsilon_j\sim N(0,0.09)\) independently of the state process. Find the predictive distribution of \(Y_j\).
The integrating-factor solution over an interval of length \(\Delta\) is \(X_{t_j}=e^{-0.5\Delta}X_{t_{j-1}}+0.4\int_{t_{j-1}}^{t_j}e^{-0.5(t_j-s)}\,dW_s\). The stochastic integral is independent of the previous state and is Gaussian with mean zero. Its variance is \(0.16\int_0^\Delta e^{-u}\,du=0.16(1-e^{-\Delta})\). Thus the transition has conditional mean \(e^{-0.5\Delta}X_{t_{j-1}}\) and conditional variance \(0.16(1-e^{-\Delta})\).
At \(\Delta=1\), the predicted state mean is \(e^{-0.5}(1)=0.60653\). Its variance includes the propagated previous-state variance and the new SDE variation:
Adding independent assay error leaves the mean unchanged and adds \(0.09\) to the variance. Therefore \(Y_j\mid\mathcal H_{j-1}\sim N(0.60653,0.28311)\). For example, the predictive density at \(y_j=0.9\) is \(\exp[-(0.9-0.60653)^2/(2(0.28311))]/\sqrt{2\pi(0.28311)}\approx0.644\). The state uncertainty contributes \(0.19311\) to the variance, while measurement error contributes \(0.09\).
Worked Example: Mixture Prediction From Two State Components
Suppose the SDE prediction gives two equally weighted components for \(X_{t_j}\): \(N(0.8,0.03)\) and \(N(1.6,0.03)\). The observation is \(Y_j=X_{t_j}+\epsilon_j\), with independent \(\epsilon_j\sim N(0,0.02)\). Find the predictive density at \(y_j=1.0\), and its mean and variance.
Adding the observation error variance to each component variance gives the observation mixture \(0.5N(0.8,0.05)+0.5N(1.6,0.05)\). Using the normal density with variance as its second parameter,
The mixture density is the weighted sum, not the average of component means: \(p(1.0\mid\mathcal H_{j-1})=0.5(1.1959)+0.5(0.04875)\approx0.6223\). The predictive mean is \(0.5(0.8)+0.5(1.6)=1.2\). For the variance, the within-component variance is \(0.05\), and the variance of the two component means is \(0.5(0.8-1.2)^2+0.5(1.6-1.2)^2=0.16\). Hence the total predictive variance is \(0.05+0.16=0.21\). A single normal distribution with this mean and variance has the same first two moments, but it does not in general have the same density as this mixture.
Worked Example: Nonlinear Concentration Mean and Separate Error Sources
Suppose the predicted state is \(0\) or \(2\), each with probability \(0.5\). The observation model is \(Y_j=e^{X_{t_j}}+\epsilon_j\), where independent assay error has mean zero and variance \(0.04\). Calculate the predictive mean and variance.
The observation mean conditional on state is \(e^{X_{t_j}}\). Therefore \(E[Y_j\mid\mathcal H_{j-1}]=0.5e^0+0.5e^2=(1+7.389056)/2=4.194528\), or approximately \(4.1945\).
The state-induced variance is the variance of the two possible conditional means, \(1\) and \(7.389056\). For two equally likely values it is one quarter of their squared difference:
The conditional measurement variance is \(0.04\) at either state, so its expected value is also \(0.04\). The total predictive variance is \(10.2050+0.04=10.2450\). Measurement error is small here, but the predictive distribution remains highly variable because uncertainty about the state is transformed nonlinearly by \(e^x\).
Parameters, Diagnostics, and Limits
The formulas above condition on fixed parameters \(\theta\). If parameters or subject-specific random effects are uncertain, a full predictive distribution averages over their conditional distribution as well. For example, if \(\eta\) denotes a subject’s random effects, the calculation integrates the transition and observation distributions over the joint filtered distribution of the current state and \(\eta\), conditional on the data. Replacing \(\eta\) and \(\theta\) by point estimates is a plug-in prediction: it can be useful, but it omits uncertainty in those quantities. In mixed-effects SDE models, this distinction matters because the same uncertainty can affect both the transition and the observation mean.
A predictive distribution can be evaluated analytically for some linear Gaussian models. Otherwise, numerical integration, simulation, or weighted particles can approximate it. With particles \(x_{j-1}^{(i)}\) and normalized weights \(w_i\), one may simulate or integrate the transition from each particle and average the resulting observation densities using those weights. The weights must represent the filtered distribution, and the simulated transition must include the interval’s known inputs. An unweighted average is justified only when the particles have equal weights.
Useful checks include verifying that the predictive density integrates to one, that all component variances are nonnegative, and that the predictive mean and variance agree with the conditional-moment identities when those moments can be calculated independently. In simulation, compare repeated simulated observations with the claimed moments. Large discrepancies can signal a missing measurement-error term, a variance-versus-standard-deviation mix-up, an incorrectly handled dose time, or a numerical approximation with too few particles. A density that assigns very little probability near many observed concentrations points to a possible mismatch in the transition, observation model, or parameter values; it does not by itself identify which part is responsible.
A common mistake is to report only the observation error variance as the predictive variance. That is correct only if the conditional observation mean is known without uncertainty, or if it is constant over the predicted state distribution. In general, the variance identity shows a second contribution from uncertainty in the predicted state. The reverse mistake is also possible: treating the variance of predicted states as the full observation variance omits assay error. The linear Gaussian example separates both terms explicitly.
The main alternatives answer different questions. A plug-in state forecast applies the observation model to a single predicted state and suppresses state uncertainty. A filtered estimate uses the current observation and therefore is not a one-step-ahead prediction. A predictive density retains the distribution of possible observations, which is useful for assessing how surprising a new measurement is. Its limitation is that a good one-step marginal prediction does not alone establish that the full SDE model is adequate; it is conditional on the specified transition, observation model, inputs, and parameter treatment.
Check Your Understanding
Answer each question, then check the worked solutions below.
- If the predicted observation-time state is \(N(3,0.16)\) and \(Y=X+\epsilon\) with independent \(\epsilon\sim N(0,0.09)\), what is the predictive distribution of \(Y\)?
- A state prediction has two equally likely values, 1 and 3. The observation is \(Y=X+\epsilon\), where the independent error variance is \(0.25\). Find the predictive mean and variance.
- Why is evaluating the observation density at the mean predicted state generally not the same as calculating the predictive density?
- If a prediction conditions on a point estimate of a random effect, which uncertainty is not represented by that conditioning choice?
Solutions. (1) The mean is \(3\); independent variances add, so the predictive variance is \(0.16+0.09=0.25\). Thus \(Y\sim N(3,0.25)\). (2) The mean is \((1+3)/2=2\). State-induced variance is \(0.5(1-2)^2+0.5(3-2)^2=1\); adding measurement variance gives \(1.25\). (3) The predictive density averages the observation density over all possible predicted states. In general, a nonlinear density evaluated at an average state is not equal to that average of densities. (4) The prediction omits uncertainty in the random effect itself, including its induced uncertainty in the transition and observation distribution.