We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy.

We investigate a novel modeling approach for end-to-end neural network training using hidden Markov models (HMM) where the transition probabilities between hidden states are modeled and learned explicitly. Most contemporary sequence-to-sequence models allow for from-scratch training by summing over all possible label segmentations in a given topology. In our approach there are explicit, learnable probabilities for transitions between segments as opposed to a blank label that implicitly encodes duration statistics. We implement a GPU-based forward-backward algorithm that enables the simultaneous training of label and transition probabilities. We investigate recognition results and additionally Viterbi alignments of our models. We find that while the transition model training does not improve recognition performance, it has a positive impact on the alignment quality. The generated alignments are shown to be viable targets in state-of-the-art Viterbi trainings.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy. Furthermore, VT converges faster and results in sparser (simpler) models, thus realizing an automatic Occam's razor for HMM learning. For more general scenario VT can be worse compared to ML but still capable of correctly recovering most of the parameters.

VT (Viterbi training), or hard EM, is an efficient way of parameter learning for probabilistic models with hidden variables. Given an observation $y$, it searches for a state of hidden variables $x$ that maximizes $p(x,y \mid \theta)$ by coordinate ascent on parameters $\theta$ and $x$. In this paper we introduce VT to PRISM, a logic-based probabilistic modeling system for generative models. VT improves PRISM in three ways. First VT in PRISM converges faster than EM in PRISM due to the VT's termination condition. Second, parameters learned by VT often show good prediction performance compared to those learned by EM. We conducted two parsing experiments with probabilistic grammars while learning parameters by a variety of inference methods, i.e.\ VT, EM, MAP and VB. The result is that VT achieved the best parsing accuracy among them in both experiments. Also we conducted a similar experiment for classification tasks where a hidden variable is not a prediction target unlike probabilistic grammars. We found that in such a case VT does not necessarily yield superior performance. Third since VT always deals with a single probability of a single explanation, Viterbi explanation, the exclusiveness condition that is imposed on PRISM programs is no more required if we learn parameters by VT. Last but not least we can say that as VT in PRISM is general and applicable to any PRISM program, it largely reduces the need for the user to develop a specific VT algorithm for a specific model. Furthermore since VT in PRISM can be used just by setting a PRISM flag appropriately, it makes VT easily accessible to (probabilistic) logic programmers. To appear in Theory and Practice of Logic Programming (TPLP).

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy. Furthermore, VT converges faster and results in sparser (simpler) models, thus realizing an automatic Occam's razor for HMM learning. For more general scenario VT can be worse compared to ML but still capable of correctly recovering most of the parameters.

Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge 1}$, observations $X_1,X_2,...$ are assumed to be conditionally independent given an ``explanatory'' Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs.