Ensemble Kalman filter log marginal likelihood.

The Ensemble Kalman Filter is a Monte Carlo version of the traditional Kalman Filter. See also [2]. It assumes the model

X[t] ~ transition_fn(X[t-1])
Y[t] ~ observation_fn(X[t])

This method estimates (logarithm of) the marginal likelihood of the observation at step t, Y[t], given state. Typically, state is the predictive ensemble at time t. In that case, this function approximates Log[p(Y[t] | Y[t-1], Y[t-2],...)] The approximation is correct under a Linear Gaussian state space model assumption, as ensemble size --> infinity.

state Instance of EnsembleKalmanFilterState at step k, conditioned on previous observations Y_{1:k}. Typically this is the output of ensemble_kalman_filter_predict.
observation Tensor representing the observation at step k.
observation_fn callable returning an instance of tfd.MultivariateNormalLinearOperator along with an extra information to be returned in the EnsembleKalmanFilterState.
perturbed_observations Whether the marginal distribution p(Y[t] | ...) is estimated using samples from the observation_fn's distribution. If False, the distribution's covariance matrix is used directly. This latter choice is less common in the literature, but works even if the ensemble size is smaller than the number of observations.
low_rank_ensemble Whether to use a LinearOperatorLowRankUpdate (rather than a dense Tensor) to represent the observation covariance. The "low rank" is the ensemble size. This is useful only if (i) the ensemble size is much less than the number of observations, and (ii) the LinearOperator associated with the observation_fn has an efficient inverse
seed PRNG seed; see tfp.random.sanitize_seed for details.
name Python str name for ops created by this method. Default value: None (i.e., 'ensemble_kalman_filter_log_marginal_likelihood').

log_marginal_likelihood Tensor with same dtype as state.

InsufficientEnsembleSizeError If perturbed_observations=True and the ensemble size is not at least one greater than the number of observations.


[1] Geir Evensen. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geophysical Research, 1994.

[2] Matthias Katzfuss, Jonathan R. Stroud & Christopher K. Wikle Understanding the Ensemble Kalman Filter. The Americal Statistician, 2016.