Parameter Estimation in Pair‐hidden Markov Models

Abstract

This paper deals with parameter estimation in pair-hidden Markov models. We first provide a rigorous formalism for these models and discuss possible definitions of likelihoods. The model is biologically motivated and therefore naturally leads to restrictions on the parameter space. Existence of two different information divergence rates is established and a divergence property is shown under additional assumptions. This yields consistency for the parameter in parametrization schemes for which the divergence property holds. Simulations illustrate different cases which are not covered by our results. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..

A Context Dependent Pair Hidden Markov Model for Statistical Alignment

“…Note that the indexes Ns, Ms of the observations generated at time s are random. This is in sharp contrast with the classical hidden Markov model (HMM) and is specific to pair-HMMs (see Arribas-Gil et al, 2006, for more details).…”

“…This criterion plays the role of a log-likelihood in the pair-HMM. (For a discussion on the quantities playing the role of likelihoods in pair-HMMs, see Arribas-Gil et al, 2006). For any integers n, m 1 and any observations X1:n and Y1:m, let wn,m(θ) := log Q θ (X1:n, Y1:m) := log P θ (∃s 1, Zs = (n, m); X1:n, Y1:m).…”

“…Statistical alignment is performed under the convenient pair-hidden Markov model, hereafter pair-HMM (see Arribas-Gil et al, 2006;Durbin et al, 1998), enabling the use of generalizations of the expectation-maximization (em) algorithm (Dempster et al, 1977). Some drawbacks of the preliminary TKF91 proposal have first been improved by the same authors in what is called the TKF92 version of the model (Thorne et al, 1992).…”

A time warping approach to multiple sequence alignment

“…. , n i , we compute the coordinates Z 1 (u),Z 2 (u) from Equation (2). We update the quantity Path(i) that stores the alignment of sequence S i to ancestor A up to position u.…”

“…The standard computational formulation of both tasks is to maximize a scoring function obtained as the sum of the score for each aligned pair of residues (nucleotides or amino acids, the highest scores being attributed to pairs of residues with highest similarity), minus some gaps penalties. Since these seminal works, an abundant literature has flourished exploring this topic in many different directions, from the pairwise problem to the more complex task of aligning more than 3 sequences [one of the very first attempts appearing in 15], from exact solutions that scale exponentially with sequence lengths to faster heuristic approaches used in the most common tools, and from the scoring formulation of the alignment problem that requires to choose the scoring parameters to probabilistic formulations in which those parameters are estimated [2,6]. However, manually refined alignments continue to be superior to purely automated methods and there is a continuous effort to improve the accuracy of MSA tools [8].…”