## Statistics Seminar: Omiros Papaspiliopoulos

**Omiros Papaspiliopoulos**, Universitat Pompeu Fabra, Barcelona

**Friday 16 May 2014, 1400-1500
A54, Postgraduate Statistics Centre Lecture Theatre**

#### Exact filtering in HMMs & the dual process

It is almost common knowledge within Statistics and several other scientific fields that in certain hidden Markov models statistical inference (filtering, smoothing, likelihood computations) can be performed without Monte Carlo methods using efficient computational algorithms that scale linearly in the length of the time series of interest; examples of such algorithms are the Kalman filter and the Baum-Welch filter. The question we will try to answer in this talk is the following: is there some fundamental structure ``hidden'' in the HMM that can be exploited to build such algorithms?

We will answer this question by using tools which are new to the filtering framework, but have been used in certain areas of Probability, among which in population genetics. In particular, we will link the solution to the filtering problem to the existence of a so-called dual process. We will show that when such process exists the filtering recursion evolves in mixtures of parametric families whose parameters can be explicitly and recursively computed. Special instance of such algorithm is the Kalman filter, but new algorithms can be devised for non-Gaussian/non-linear models, for example for models where the signal is the well-known CIR process. We discuss application of these ideas to HMMs for which the underlying signal is a probability distribution. Finally, we provide some results about the numerical complexity of this framework.