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Estimation of Viterbi Path in Bayesian Hidden Markov Models
Estimation of Viterbi path in Bayesian hidden Markov models Jüri Lember1,∗ Dario Gasbarra2, Alexey Koloydenko3 and Kristi Kuljus1 May 14, 2019 1University of Tartu, Estonia; 2University of Helsinki, Finland; 3Royal Holloway, University of London, UK Abstract The article studies different methods for estimating the Viterbi path in the Bayesian framework. The Viterbi path is an estimate of the underlying state path in hidden Markov models (HMMs), which has a maximum joint posterior probability. Hence it is also called the maximum a posteriori (MAP) path. For an HMM with given parameters, the Viterbi path can be easily found with the Viterbi algorithm. In the Bayesian framework the Viterbi algorithm is not applicable and several iterative methods can be used instead. We introduce a new EM-type algorithm for finding the MAP path and compare it with various other methods for finding the MAP path, including the variational Bayes approach and MCMC methods. Examples with simulated data are used to compare the performance of the methods. The main focus is on non-stochastic iterative methods and our results show that the best of those methods work as well or better than the best MCMC methods. Our results demonstrate that when the primary goal is segmentation, then it is more reasonable to perform segmentation directly by considering the transition and emission parameters as nuisance parameters. Keywords: HMM, Bayes inference, MAP path, Viterbi algorithm, segmentation, EM, variational Bayes, simulated annealing 1 Introduction and preliminaries arXiv:1802.01630v2 [stat.CO] 11 May 2019 Hidden Markov models (HMMs) are widely used in application areas including speech recognition, com- putational linguistics, computational molecular biology, and many more. -
Introduction to Machine Learning Lecture 3
Introduction to Machine Learning Lecture 3 Mehryar Mohri Courant Institute and Google Research [email protected] Bayesian Learning Bayes’ Formula/Rule Terminology: likelihood prior Pr[X Y ]Pr[Y ] Pr[Y X]= | . posterior| Pr[X] probability evidence Mehryar Mohri - Introduction to Machine Learning page 3 Loss Function Definition: function L : R + indicating the penalty for an incorrectY prediction.×Y→ • L ( y,y ) : loss for prediction of y instead of y . Examples: • zero-one loss: standard loss function in classification; L ( y,y )=1 y = y for y,y . ∈Y • non-symmetric losses: e.g., for spam classification; L ( ham , spam) L ( spam , ham) . ≤ squared loss: standard loss function in • 2 regression; L ( y,y )=( y y ) . − Mehryar Mohri - Introduction to Machine Learning page 4 Classification Problem Input space : e.g., set of documents. X N feature vector Φ ( x ) R associated to x . • ∈ ∈X N notation: feature vector x R . • ∈ • example: vector of word counts in document. Output or target space : set of classes; e.g., sport, Y business, art. Problem: given x , predict the correct class y ∈Y associated to x . Mehryar Mohri - Introduction to Machine Learning page 5 Bayesian Prediction Definition: the expected conditional loss of predicting y is ∈Y [y x]= L(y,y)Pr[y x]. L | | y ∈Y Bayesian decision : predict class minimizing expected conditional loss, that is y∗ =argmin [y x]=argmin L(y,y)Pr[y x]. L | | y y y ∈Y zero-oneb loss: y∗ =argmaxb Pr[y x]. • y | Maximum a Posteriori (MAP) principle. b Mehryar Mohri - Introduction to Machine Learning page 6 Binary Classification - Illustration 1 Pr[y x] 1 | Pr[y x] 2 | 0 x Mehryar Mohri - Introduction to Machine Learning page 7 Maximum a Posteriori (MAP) Definition: the MAP principle consists of predicting according to the rule y =argmaxPr[y x]. -
Additive Smoothing for Relevance-Based Language Modelling of Recommender Systems
Additive Smoothing for Relevance-Based Language Modelling of Recommender Systems Daniel Valcarce, Javier Parapar, Álvaro Barreiro Information Retrieval Lab Department of Computer Science University of A Coruña, Spain {daniel.valcarce, javierparapar, barreiro}@udc.es ABSTRACT the users' past behaviour. Continuous developments in this The use of Relevance-Based Language Models for top-N rec- field have been made to meet these high expectations. ommendation has become a promising line of research. Pre- We can distinguish multiple approaches to recommenda- vious works have used collection-based smoothing methods tion [24]. They are often classified in three main categories: for this task. However, a recent analysis on RM1 (an estima- content-based, collaborative filtering and hybrid techniques. tion of Relevance-Based Language Models) in document re- Content-based approaches generate recommendations based trieval showed that this type of smoothing methods demote on the item and user descriptions: they suggest items simi- the IDF effect in pseudo-relevance feedback. In this paper, lar to those liked by the target user [9]. In contrast, collab- we claim that the IDF effect from retrieval is closely related orative filtering methods rely on the interactions (typically to the concept of novelty in recommendation. We perform ratings) between users and items in the system [21]. Finally, an axiomatic analysis of the IDF effect on RM2 conclud- there exist hybrid algorithms that combine both collabora- ing that this kind of smoothing methods also demotes the tive filtering and content-based approaches. IDF effect in recommendation. By axiomatic analysis, we Traditionally, Information Retrieval (IR) has focused on find that a collection-agnostic method, Additive smoothing, delivering the information that users demand [1]. -
Smoothing Parameter and Model Selection for General Smooth Models
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION , VOL. , NO. , –, Theory and Methods http://dx.doi.org/./.. Smoothing Parameter and Model Selection for General Smooth Models Simon N. Wooda,NatalyaPyab, and Benjamin Säfkenc aSchool of Mathematics, University of Bristol, Bristol, UK; bSchool of Science and Technology, Nazarbayev University, Astana, Kazakhstan, and KIMEP University, Almaty, Kazakhstan; cChairs of Statistics and Econometrics, Georg-August-Universität Göttingen, Germany ABSTRACT ARTICLE HISTORY This article discusses a general framework for smoothing parameter estimation for models with regular like- Received October lihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and Revised March parametric terms may also be present. By construction the method is numerically stable and convergent, KEYWORDS and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known Additive model; AIC; problem with AIC for such models, thereby improving the range of model selection tools available. The Distributional regression; smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penal- GAM; Location scale and ties measuring function smoothness. Model estimation is by penalized likelihood maximization, where shape model; Ordered the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate categorical regression; marginal likelihood. The methods cover, for example, generalized additive models for nonexponential fam- Penalized regression spline; ily responses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distribu- REML; Smooth Cox model; tions), generalized additive models for location scale and shape (e.g., two stage zero inflation models, and Smoothing parameter uncertainty; Statistical Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. -
Introduction to Naivebayes Package
Introduction to naivebayes package Michal Majka March 8, 2020 1 Introduction The naivebayes package provides an efficient implementation of the popular Naïve Bayes classifier. It was developed and is now maintained based on three principles: it should be efficient, user friendly and written in Base R. The last implies no dependencies, however, it neither denies nor interferes with the first as many functions from the Base R distribution use highly efficient routines programmed in lower level languages, such as C or FORTRAN. In fact, the naivebayes package utilizes only such functions for resource-intensive calculations. This vignette should make the implementation of the general naive_bayes() function more transparent and give an overview over its functionalities. 2 Installation Just like many other R packages, the naivebayes can be installed from the CRAN repository by simply typing into the console the following line: install.packages("naivebayes") An alternative way of obtaining the package is first downloading the package source from https://CRAN.R-project.org/package=naivebayes, specifying the location of the file and running in the console: # path_to_tar.gz file <- " " install.packages(path_to_tar.gz, repos= NULL, type= "source") The full source code can be viewed either on the official GitHub CRAN repository: https: //github.com/cran/naivebayes or on the development repository: https://github.com/ majkamichal/naivebayes. After successful installation, the package can be used with: library(naivebayes) 3 Main functions The general function -
Adding Improvements to Multinomial Naive Bayes for Increasing the Accuracy of Aggressive Tweets Classification
ISSN (Online) 2278-1021 IJARCCE ISSN (Print) 2319-5940 International Journal of Advanced Research in Computer and Communication Engineering Vol. 8, Issue 11, November 2019 Adding Improvements to Multinomial Naive Bayes for Increasing the Accuracy of Aggressive Tweets Classification Divisha Bisht1, Sanjay Joshi2 Student, Department of Information Technology, College of Technology, GBPUA&T, Pantnagar, India1 Assistant Professor, Department of Information Technology, College of Technology, GBPUA&T, Pantnagar, India2 Abstract: Naïve Bayes is a popular supervised learning method widely used for text classification and sentiment analysis. There has been a rise of aggressive troll comments in the social networking sites which leads to online harassment and causes distressful online experiences. This paper uses Naïve Bayes classifier using Bag of Words on ‘Tweets dataset for Detection of Cyber-Trolls’ (dataset taken from Kaggle) and aims to improve baseline model by adding cumulative changes and studying their impact on the performance of the model. Keywords: Naïve Bayes, Classification, Improvements, Accuracy I. INTRODUCTION The prevalence, ease of access and anonymity in the social media communities supported by the all dominating web of internet has led to a rise of online negative behaviour in the form of trolling. A regular influx of a large number of troll comments can be observed in most Social Networking Sites (SNS) like Twitter, Facebook, Reddit. The Collins English Dictionary defines troll as “someone who posts unkind or offensive messages -
Viterbi Extraction Tutorial with Hidden Markov Toolkit
Viterbi Extraction tutorial with Hidden Markov Toolkit Zulkarnaen Hatala 1, Victor Puturuhu 2 Electrical Engineering Department Politeknik Negeri Ambon Ambon, Indonesia [email protected] [email protected] Abstract —An algorithm used to extract HMM parameters condition probabilities. The distribution of observation only is revisited. Most parts of the extraction process are taken from depends on the current state of Markov chain. In some implemented Hidden Markov Toolkit (HTK) program under application not all states of Markov chain emit observations. name HInit. The algorithm itself shows a few variations Set of states that emit an observation is called emitting states, compared to another domain of implementations. The HMM while non emitting states could be an entry or start state, model is introduced briefly based on the theory of Discrete absorption or exiting state and intermediate or delaying state. Time Markov Chain. We schematically outline the Viterbi The entry state is the Markov chain state with initial method implemented in HTK. Iterative definition of the probability 1. method which is ready to be implemented in computer programs is reviewed. We also illustrate the method One application of Hidden Markov Model is in speech calculation precisely using manual calculation and extensive recognition. The observations of HMM are used to model graphical illustration. The distribution of observation sequence of speech feature vectors like Mel Frequency probability used is simply independent Gaussians r.v.s. The Cepstral Coefficients (MFCC) [4]. MFCC is assumed to be purpose of the content is not to justify the performance or generated by emitting hidden states of the underlying accuracy of the method applied in a specific area. -
Markov Chains and Hidden Markov Models
COMP 182 Algorithmic Thinking Luay Nakhleh Markov Chains and Computer Science Hidden Markov Models Rice University ❖ What is p(01110000)? ❖ Assume: ❖ 8 independent Bernoulli trials with success probability of α? ❖ Answer: ❖ (1-α)5α3 ❖ However, what if the assumption of independence doesn’t hold? ❖ That is, what if the outcome in a Bernoulli trial depends on the outcomes of the trials the preceded it? Markov Chains ❖ Given a sequence of observations X1,X2,…,XT ❖ The basic idea behind a Markov chain (or, Markov model) is to assume that Xt captures all the relevant information for predicting the future. ❖ In this case: T p(X1X2 ...XT )=p(X1)p(X2 X1)p(X3 X2) p(XT XT 1)=p(X1) p(Xt Xt 1) | | ··· | − | − t=2 Y Markov Chains ❖ When Xt is discrete, so Xt∈{1,…,K}, the conditional distribution p(Xt|Xt-1) can be written as a K×K matrix, known as the transition matrix A, where Aij=p(Xt=j|Xt-1=i) is the probability of going from state i to state j. ❖ Each row of the matrix sums to one, so this is called a stochastic matrix. 590 Chapter 17. Markov and hidden Markov models Markov Chains 1 α 1 β − α − A11 A22 A33 1 2 A12 A23 1 2 3 β ❖ A finite-state Markov chain is equivalent(a) (b) Figure 17.1 State transition diagrams for some simple Markov chains. Left: a 2-state chain. Right: a to a stochastic automaton3-state left-to-right. chain. ❖ One way to represent aAstationary,finite-stateMarkovchainisequivalenttoa Markov chain is stochastic automaton.Itiscommon 590 to visualizeChapter such 17. -
N-Best Hypotheses
FIANAL PROJECT REPORT NAME: Shuan wang STUDENT #: 21224043 N-best Hypotheses [INDEX] 1. Introduction 1.1 Project Goal 1.2 Sources of Interest 1.3 Current Work 2. Implemented Three Algorithms 2.1 General Viterbi Algorithm 2.1.1 Basic Ideas 2.1.2 Implementation 2.2 N-Best List in HMM 2.2.1 Basic Ideas 2.2.2 Implementation 2.3 BMMF (Best Max-Marginal First) 2.3.1 Basic Ideas 2.3.2 Implementation 3. Some Experiment Results 4. Future Work 5. References 1 of 8 FIANAL PROJECT REPORT NAME: Shuan wang STUDENT #: 21224043 1. Introduction It’s very useful to find the top N most probable figures of hidden variable sequence. There are some important methods in recent years to find such kind of N figures: (1) General Viterbi Algorithm (2) MFP (Max-Flow Propagation), by Nilson and Lauritzen in 1998 (3) N-Best List in HMM, by Nilson and Goldberger in 2001 (4) BMMF (Best Max-Marginal First), by Chen Yanover and Yair Weiss in 2003 In this project my task is to realize part of these algorithms and test the results. 1.1 Project Goal The project goal is to totally understand the basic algorithm ideas and try to realize the most recent three of them (MFP, N-Best List in HMM, BMMF) and test them. 1.2 Sources of Interest Many applications, such as protein folding, vocabulary speech recognition and image analysis, want to find not just the best configuration of the hidden variables (state sequence), but rather the top N. Normally the recognizers in these applications are based on a relatively simple model. -
Efficient Learning of Smooth Probability Functions from Bernoulli Tests With
Efficient learning of smooth probability functions from Bernoulli tests with guarantees Paul Rolland 1 Ali Kavis 1 Alex Immer 1 Adish Singla 2 Volkan Cevher 1 Abstract grows cubicly with the number of tests, and can thus be inapplicable when the amount of data becomes large. There We study the fundamental problem of learning an has been extensive work to resolve this cubic complexity unknown, smooth probability function via point- associated with GP computations (Rasmussen, 2004). How- wise Bernoulli tests. We provide a scalable algo- ever, these methods require additional approximations on rithm for efficiently solving this problem with rig- the posterior distribution, which impacts the efficiency and orous guarantees. In particular, we prove the con- make the overall algorithm even more complicated, leading vergence rate of our posterior update rule to the to further difficulties in establishing theoretical convergence true probability function in L2-norm. Moreover, guarantees. we allow the Bernoulli tests to depend on con- Recently, Goetschalckx et al.(2011) tackled the issues en- textual features and provide a modified inference countered by LGP, and proposed a scalable inference engine engine with provable guarantees for this novel based on Beta Processes called Continuous Correlated Beta setting. Numerical results show that the empirical Process (CCBP) for approximating the probability function. convergence rates match the theory, and illustrate By scalable, we mean the algorithm complexity scales lin- the superiority of our approach in handling con- early with the number of tests. However, no theoretical anal- textual features over the state-of-the-art. ysis is provided, and the approximation error saturates as the number of tests becomes large (cf., section 5.1). -
Bias/Variance Tradeoff
Behavioral Data Mining Lecture 2 Outline Review of Statistical Learning • Linear Regression, Nearest Neighbor Estimator, Loss • Bias-Variance Tradeoff • Naïve Bayes Classifier Statistical Learning We will follow the exposition in Hastie et al., “Statistical Learning” Variables: Qualitative (discrete): {True, False}, {Red, Green, Blue}, {1, 2, 3, 4, 5}, or latent (anonymous) factors like {F1, F2, F3, F4, F5} Quantitative (continuous): Real values ∈ ℝ, or values in an interval. e.g. height, weight, frequency,… Statistical Learning Variables will often be vectors (uppercase italic) X = (X1,…, Xn) 푋11 ⋯ 푋1푛 or m x n matrices (uppercase bold) 퐗 = ⋮ ⋱ ⋮ 푋푚1 ⋯ 푋푚푛 Observations will be written as matching lowercase, e.g. a series of observations of X would be 푥1, … , 푥푝 Estimates will be written as symbols with a hat: 푌 for a variable 푌 or 훽 for a parameter 훽. Linear Regression The predicted value of Y is given by: 푝 푌 = 훽0 + 푋푗훽푗 푗=1 and the vector of coefficients 훽 comprise the regression model. In the general case, we can have a vector output and a matrix of coefficients and write 푌 = 훽퐗 (assume 푋0 = 1). (thinking of observations as columns rather than rows will work better for matrix algorithms, i.e. 퐗 is transposed rel. to book). Statistical Learning 푋11 ⋯ 푋1푛 i.e. columns of 퐗 = ⋮ ⋱ ⋮ 푋푚1 ⋯ 푋푚푛 are distinct observations, rows of X are input features. Same for output features. Residual Sum-of-Squares To determine the model parameters 훽 from some data, we need to define an error function, like Residual Sum of Squares: 푁 2 RSS 훽 = 푦푖 − 훽푥푖 푖=1 or symbolically RSS 훽 = 퐲 − 훽퐗 푇 퐲 − 훽퐗 . -
Package 'Naivebayes'
Package ‘naivebayes’ March 8, 2020 Type Package Title High Performance Implementation of the Naive Bayes Algorithm Version 0.9.7 Author Michal Majka Maintainer Michal Majka <[email protected]> Description In this implementation of the Naive Bayes classifier following class conditional distribu- tions are available: Bernoulli, Categorical, Gaussian, Poisson and non-parametric representa- tion of the class conditional density estimated via Kernel Density Estimation. Implemented clas- sifiers handle missing data and can take advantage of sparse data. URL https://github.com/majkamichal/naivebayes, https://majkamichal.github.io/naivebayes/ BugReports https://github.com/majkamichal/naivebayes/issues License GPL-2 Encoding UTF-8 NeedsCompilation no VignetteBuilder knitr Suggests knitr, Matrix RoxygenNote 6.1.1 Repository CRAN Date/Publication 2020-03-08 18:20:05 UTC R topics documented: bernoulli_naive_bayes . .2 coef .............................................5 gaussian_naive_bayes . .6 get_cond_dist . .8 Infix operators . .9 multinomial_naive_bayes . 10 naivebayes . 12 1 2 bernoulli_naive_bayes naive_bayes . 13 nonparametric_naive_bayes . 18 plot.bernoulli_naive_bayes . 20 plot.gaussian_naive_bayes . 21 plot.naive_bayes . 22 plot.nonparametric_naive_bayes . 24 plot.poisson_naive_bayes . 26 poisson_naive_bayes . 27 predict.bernoulli_naive_bayes . 30 predict.gaussian_naive_bayes . 31 predict.multinomial_naive_bayes . 33 predict.naive_bayes . 34 predict.nonparametric_naive_bayes . 36 predict.poisson_naive_bayes . 38 tables . 39 Index 41