Filtering Distributions for Nonlinear State-Space Models

Three Distributions for a Latent State

A state estimate is not always a single concentration or compartment amount. In a nonlinear state-space model, uncertainty about the latent state is represented by a distribution, and the observations included in that distribution matter. The distribution before processing the current observation is a prediction; after processing it, it is a filter. If observations from later times are also used to infer an earlier state, the result is a smoother.

Let \(Z_k\) denote the latent state at observation time \(k\), and let \(Y_k\) denote the corresponding observation. Write \(y_{1:k}=(y_1,\ldots,y_k)\) for the observed data through time \(k\). A concentration may be a nonlinear function \(h(Z_k)\) of the state, such as the exponential of a log-concentration state.

These are distributions, not three names for the same point estimate. In particular, the one-step predictive distribution of the latent state is different from the predictive distribution of the next observation, \(p(y_k\mid y_{1:k-1})\). The latter is obtained by averaging the observation model over the latent-state prediction.

Bayesian Filtering Recursion

Assume a first-order Markov state model and conditionally independent observations: given the state sequence, observation \(Y_k\) depends on the states only through \(Z_k\). Suppose the initial state has a density, the transition from \(Z_{k-1}=z_{k-1}\) to \(Z_k=z_k\) has density \(q_k(z_k\mid z_{k-1})\), and the observation has density \(g_k(y_k\mid z_k)\). Densities are taken with respect to fixed dominating measures. Assume the integrals below exist and the observation normalizing constant is finite and strictly positive. The initial distribution is the filtering distribution at time zero, before observations.

The prediction step propagates the preceding filter through the state transition. The update step weights that prediction by how likely the current observation is at each possible state.

Derivation. Given the observations through \(k-1\), the Markov property makes the conditional distribution of \(Z_k\) depend on the past state through \(Z_{k-1}\). Integrating over that preceding state gives the prediction by the law of total probability. For the update, the joint conditional density of \(Z_k\) and \(Y_k=y_k\), given \(y_{1:k-1}\), factors as \(g_k(y_k\mid z_k)p(z_k\mid y_{1:k-1})\). Dividing by its integral over \(z_k\) gives the conditional density of \(Z_k\) given \(y_{1:k}\), by Bayes' rule. The denominator is also the one-step predictive observation density \(p(y_k\mid y_{1:k-1})\); its positivity makes the update well-defined. \(\square\)

The recursion is valid for nonlinear transitions and nonlinear observation functions. Nonlinearity generally prevents the distribution from remaining in a convenient finite-dimensional family, but it does not change the definitions or the Bayesian identities. The extended Kalman filter instead propagates an approximate state summary under additional approximations; it is not, in general, the exact nonlinear filtering distribution.

Filtering Is Not Smoothing

A filter uses no observations later than the state time being estimated. A smoother does use later observations, and can consequently revise the inferred distribution of an earlier state. This distinction follows from the same factorization that underlies the filtering recursion.

Proof. Conditional on \(Z_k=z\), the next state has density \(q_{k+1}(u\mid z)\), and the next observation has density \(g_{k+1}(y_{k+1}\mid u)\). Given \(Z_{k+1}=u\), the remaining observations have likelihood \(\beta_{k+1}(u)\). Integrating their product over \(u\) gives the stated backward recursion. For the smoothing identity, factor the joint conditional density of \(Z_k\) and the future observations given \(y_{1:k}\) as \(p(z_k\mid y_{1:k})p(y_{k+1:T}\mid z_k,y_{1:k})\). The Markov property and conditional independence make the second factor \(\beta_k(z_k)\). Normalizing this product over \(z_k\) yields the conditional density given all observations through \(T\). \(\square\)

The backward factor \(\beta_k\) is large for states that make the future observations plausible. Thus smoothing is not simply filtering performed with a different numerical method: it conditions on a different data set.

Nonlinear Concentration Summaries

When the state is transformed to concentration, transform the distribution rather than assuming that a transformed mean is a mean. For a function \(h\), the distribution of \(h(Z_k)\) is the probability assigned to its values by the distribution of \(Z_k\). Its expectation is \(E[h(Z_k)\mid y_{1:k}]\), not generally \(h(E[Z_k\mid y_{1:k}])\). This matters when reporting concentration from a filtered log-state or from a vector of compartment states.

Diagnostics, Alternatives, and Limitations

For an exact density update, check that the prediction integrates to one, that the likelihood is evaluated at the correct state-dependent concentration, and that the update denominator is finite and positive. In a numerical approximation, check that reported probabilities or weights normalize to one and that posterior summaries respond in the direction implied by the likelihood. A posterior that barely changes despite a highly discriminating observation may signal a coding, scaling, or likelihood problem; it may also reflect a genuinely diffuse prediction.

The exact recursion is an integral identity, not a promise that the integrals can be evaluated analytically. Linear Gaussian models admit a compact Gaussian recursion. For nonlinear models, an extended Kalman filter uses local approximations; quadrature or grid methods approximate integrals on selected state spaces; particle methods represent distributions with samples and weights. These alternatives differ in computational cost and approximation error. The filtering distribution remains the target object, regardless of the method used to approximate it.

Approximation quality can deteriorate when the state is high-dimensional, when observations are very informative relative to the predictive distribution, or when the transition and observation models are badly misspecified. An apparently precise filtered concentration does not account for uncertainty omitted from the model. Also, predictions for a future observation must average over state uncertainty; substituting only a point estimate into a nonlinear observation function can give a misleading result.