{"title":"Local Viterbi property in decoding","authors":"J. Lember","doi":"10.1093/imaiai/iaad004","DOIUrl":null,"url":null,"abstract":"\n The article studies the decoding problem (also known as the classification or the segmentation problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and the underlying state sequence form a two-dimensional Markov chain, a natural generalization of hidden Markov model. The standard solutions to the decoding problem are the so-called Viterbi path—a sequence with maximum state path probability given the observations—or the pointwise maximum a posteriori (PMAP) path that maximizes the expected number of correctly classified entries. When the goal is to simultaneously maximize both criterions—conditional probability (corresponding to Viterbi path) and pointwise conditional probability (corresponding to PMAP path)—then they are combined into one single criterion via the regularization parameter $C$. The main objective of the article is to study the behaviour of the solution—called the hybrid path—as $C$ grows. Increasing $C$ increases the conditional probability of the hybrid path and when $C$ is big enough then every hybrid path is a Viterbi path. We show that hybrid paths also approach the Viterbi path locally: we define $m$-locally Viterbi paths and show that the hybrid path is $m$-locally Viterbi whenever $C$ is big enough. This all might lead to an impression that when $C$ is relatively big then any hybrid path that is not yet Viterbi differs from the Viterbi path by a few single entries only. We argue that this intuition is wrong, because when unique and $m$-locally Viterbi, then different hybrid paths differ by at least $m$ entries. Thus, when $C$ increases then the different hybrid paths tend to differ from each other by larger and larger intervals. Hence the hybrid paths might offer a variety of rather different solutions to the decoding problem.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imaiai/iaad004","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 1
Abstract
The article studies the decoding problem (also known as the classification or the segmentation problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and the underlying state sequence form a two-dimensional Markov chain, a natural generalization of hidden Markov model. The standard solutions to the decoding problem are the so-called Viterbi path—a sequence with maximum state path probability given the observations—or the pointwise maximum a posteriori (PMAP) path that maximizes the expected number of correctly classified entries. When the goal is to simultaneously maximize both criterions—conditional probability (corresponding to Viterbi path) and pointwise conditional probability (corresponding to PMAP path)—then they are combined into one single criterion via the regularization parameter $C$. The main objective of the article is to study the behaviour of the solution—called the hybrid path—as $C$ grows. Increasing $C$ increases the conditional probability of the hybrid path and when $C$ is big enough then every hybrid path is a Viterbi path. We show that hybrid paths also approach the Viterbi path locally: we define $m$-locally Viterbi paths and show that the hybrid path is $m$-locally Viterbi whenever $C$ is big enough. This all might lead to an impression that when $C$ is relatively big then any hybrid path that is not yet Viterbi differs from the Viterbi path by a few single entries only. We argue that this intuition is wrong, because when unique and $m$-locally Viterbi, then different hybrid paths differ by at least $m$ entries. Thus, when $C$ increases then the different hybrid paths tend to differ from each other by larger and larger intervals. Hence the hybrid paths might offer a variety of rather different solutions to the decoding problem.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.