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Exact inference for hidden Markov models requires the evaluation of all distributions of interest - filtering, prediction, smoothing and likelihood - with a finite computational effort. This article provides sufficient conditions for exact inference for a class of hidden Markov models on general state spaces given a set of discretely collected indirect observations linked non linearly to the signal, and a set of practical algorithms for inference. The conditions we obtain are concerned with the existence of a certain type of dual process, which is an auxiliary process embedded in the time reversal of the signal, that in turn allows to represent the distributions and functions of interest as finite mixtures of elementary densities or products thereof. We describe explicitly how to update recursively the parameters involved, yielding qualitatively similar results to those obtained with Baum--Welch filters on finite state spaces. We then provide practical algorithms for implementing the recursions, as well as approximations thereof via an informed pruning of the mixtures, and we show superior performance to particle filters both in accuracy and computational efficiency. The code for optimal filtering, smoothing and parameter inference is made available in the Julia package DualOptimalFiltering.
We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the correspondi
We provide a sufficient criterion for the unique parameter identification of combinatorially symmetric Hidden Markov Models based on the structure of their transition matrix. If the observed states of the chain form a zero forcing set of the graph of
Recently a new algorithm for sampling posteriors of unnormalised probability densities, called ABC Shadow, was proposed in [8]. This talk introduces a global optimisation procedure based on the ABC Shadow simulation dynamics. First the general method
Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompan
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