Suppose that we are given a time series where consecutive samples are believed to come from a probabilistic source, that the source changes from time to time and that the total number of sources is fixed. Our objective is to estimate the distributions of the sources. A standard approach to this problem is to model the data as a hidden Markov model (HMM). However, since the data often lacks the Markov or the stationarity properties of an HMM, one can ask whether this approach is still suitable or perhaps another approach is required. In this paper we show that a maximum likelihood HMM estimator can be used to approximate the source distributions in a much larger class of models than HMMs. Specifically, we propose a natural and fairly general non-stationary model of the data, where the only restriction is that the sources do not change too often. Our main result shows that for this model, a maximum-likelihood HMM estimator produces the correct second moment of the data, and the results can be extended to higher moments.
Finally, we make essential use of type theory for Markov chains. The results we use were obtained in @cite_26 , where second order and higher order type theory is developed.
@cite_26
2319462502 2136796925 2748981990 2147568833
A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain. WRIGHT-FISHER MODEL; NEUTRAL MUTATION; RANDOM EQUIVALENCE RELATIONS; COALESCENT; EWENS SAMPLING FORMULA; COUPLING; ENTRANCE BOUNDARY Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also gives guidance on the choice of sample size and allocation We formalise the axiomatic set theory second-order ZF in the constructive type theory of Coq assuming excluded middle. In this setting we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the correspondence of inner models and Grothendieck universes. Our results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and transfinite recursion. We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.