Partially exchangeable Hidden Markov Models

Countable Partially Exchangeable Mixtures

“…By the same token, the characterization of countable mixtures of Markov chains of order k holds true, for the proof see [10]. Unfortunately, the approach of [7] and [10] does not readily generalize to Polish state space. Based on the results in [8], a de Finetti's type representation theorem for mixtures of semi-Markov processes have been proved in [6].…”

“…A characterization of countable mixtures of Markov chains can be given also in the setup of [4], using different mathematical tools. The result is in [7], but for a complete proof see [10]. By the same token the characterization of countable mixtures of Markov chains of order k holds true, for the proof see [10].…”

“…By the same token the characterization of countable mixtures of Markov chains of order k holds true, for the proof see [10]. Unfortunately the approach of [7] and [10] does not readily generalize to Polish state space.…”

“…For the converse assume that (Y n ) is a countable mixture of Markov chains, with random kernel t taking values t h h∈H and with µ h = P( t = t h ), and finite distributions as in Equation (7). To prove that (Y n ) is a HMM with recurrent underlying Markov chain we construct a recurrent Markov chain (X n ) and a sequence ( Y n ) satisfying the first two conditions in Definition 2.8, then showing that ( Y n ) has the same distributions of (Y n ) 4 .…”

“…Lemma 6.10. Consider a HMM (Y n ) where the underlying Markov chain (X n ) has transition matrix P as in (7), invariant measure π as in (8), and assigned read-out distributions f x ( S), then (Y n ) is a countable mixtures of i.i.d. sequences where p takes values in the set {F h , h ∈ H}, with…”