Modelling regimes with Bayesian network mixtures Supplementary material
نویسندگان
چکیده
1 Parameter estimation Since we cannot observe the hidden variablesH1:T , we cannot solve the parameter estimation problem by simply counting events from a dataset. Instead we must apply some method that can give us an approximate solution. The canonical way of parameter estimation in HMM is to use EM, and therefore we shall also adopt this technique for our GBN-HMMs. As before, let o1:T represent a sequence of observations over the variablesO1:T and let h1:T = {h1, h2, ..., hT } represent a sequence of states. Let H represent the set of all state sequences h1:T . The current parameters for our model are denoted Θ′, and we seek parameters Θ that maximises the expected log-likelihood of the observed data. This expectation is expressed by Q(Θ,Θ′) = ∑ h1:T∈H p(o1:T , h1:T |Θ ′) log p(o1:T , h1:T |Θ), thus it is the function Q that we wish to maximise. We can substitute p(o1:T , h1:T |Θ) in the Q function with our factorisation of the GBNHMM, which gives us the expanded Q function in Equation 1. From this expansion we can see that the terms do not interact, thus we can maximise each term separately.
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