DOCSLIB.ORG

Cube Summing, Approximate Inference with Non-Local Features, and Dynamic Programming Without Semirings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cube Summing, Approximate Inference with Non-Local Features, and Dynamic Programming Without Semirings Cube Summing, Approximate Inference with Non-Local Features, and Dynamic Programming without Semirings Kevin Gimpel and Noah A. Smith Language Technologies Institute Carnegie Mellon University Pittsburgh, PA 15213, USA {kgimpel,nasmith}@cs.cmu.edu Abstract features, but leaves open the question of how the feature weights or probabilities are learned. Mean- We introduce cube summing, a technique while, some learning algorithms, like maximum that permits dynamic programming algo- likelihood for conditional log-linear models (Laf- rithms for summing over structures (like ferty et al., 2001), unsupervised models (Pereira the forward and inside algorithms) to be and Schabes, 1992), and models with hidden vari- extended with non-local features that vio- ables (Koo and Collins, 2005; Wang et al., 2007; late the classical structural independence Blunsom et al., 2008), require summing over the assumptions. It is inspired by cube prun- scores of many structures to calculate marginals. ing (Chiang, 2007; Huang and Chiang, We first review the semiring-weighted logic 2007) in its computation of non-local programming view of dynamic programming al- features dynamically using scored k-best gorithms (Shieber et al., 1995) and identify an in- lists, but also maintains additional resid- tuitive property of a program called proof locality ual quantities used in calculating approx- that follows from feature locality in the underlying imate marginals. When restricted to lo- probability model (§2). We then provide an analy- cal features, cube summing reduces to a sis of cube pruning as an approximation to the in- novel semiring (k-best+residual) that gen- tractable problem of exact optimization over struc- eralizes many of the semirings of Good- tures with non-local features and show how the man (1999). When non-local features are use of non-local features with k-best lists breaks included, cube summing does not reduce certain semiring properties (§3). The primary to any semiring, but is compatible with contribution of this paper is a novel technique— generic techniques for solving dynamic cube summing—for approximate summing over programming equations. discrete structures with non-local features, which we relate to cube pruning (§4). We discuss imple- 1 Introduction mentation (§5) and show that cube summing be- Probabilistic NLP researchers frequently make in- comes exact and expressible as a semiring when dependence assumptions to keep inference algo- restricted to local features; this semiring general- rithms tractable. Doing so limits the features that izes many commonly-used semirings in dynamic are available to our models, requiring features programming (§6). to be structurally local. Yet many problems in 2 Background NLP—machine translation, parsing, named-entity recognition, and others—have benefited from the In this section, we discuss dynamic programming addition of non-local features that break classical algorithms as semiring-weighted logic programs. independence assumptions. Doing so has required We then review the definition of semirings and im- algorithms for approximate inference. portant examples. We discuss the relationship be- Recently cube pruning (Chiang, 2007; Huang tween locally-factored structure scores and proofs and Chiang, 2007) was proposed as a way to lever- in logic programs. age existing dynamic programming algorithms that find optimal-scoring derivations or structures 2.1 Dynamic Programming when only local features are involved. Cube prun- Many algorithms in NLP involve dynamic pro- ing permits approximate decoding with non-local gramming (e.g., the Viterbi, forward-backward, probabilistic Earley’s, and minimum edit distance Semirings define these values and define two op- algorithms). Dynamic programming (DP) in- erators over them, called “aggregation” (max in volves solving certain kinds of recursive equations Eq. 1) and “combination” (× in Eq. 1). with shared substructure and a topological order- Goodman and Eisner et al. assumed that the val- ing of the variables. ues of the variables are in a semiring, and that the Shieber et al. (1995) showed a connection equations are defined solely in terms of the two between DP (specifically, as used in parsing) semiring operations. We will often refer to the and logic programming, and Goodman (1999) “probability” of a proof, by which we mean a non- augmented such logic programs with semiring negative R-valued score defined by the semantics weights, giving an algebraic explanation for the of the dynamic program variables; it may not be a intuitive connections among classes of algorithms normalized probability. with the same logical structure. For example, in Goodman’s framework, the forward algorithm and 2.2 Semirings the Viterbi algorithm are comprised of the same A semiring is a tuple hA, ⊕, ⊗, 0, 1i, in which A logic program with different semirings. Goodman is a set, ⊕ : A × A → A is the aggregation defined other semirings, including ones we will operation, ⊗ : A × A → A is the combina- use here. This formal framework was the basis tion operation, 0 is the additive identity element for the Dyna programming language, which per- (∀a ∈ A, a ⊕ 0 = a), and 1 is the multiplica- mits a declarative specification of the logic pro- tive identity element (∀a ∈ A, a ⊗ 1 = a). A gram and compiles it into an efficient, agenda- semiring requires ⊕ to be associative and com- based, bottom-up procedure (Eisner et al., 2005). mutative, and ⊗ to be associative and to distribute For our purposes, a DP consists of a set of recur- over ⊕. Finally, we require a ⊗ 0 = 0 ⊗ a = 0 for sive equations over a set of indexed variables. For all a ∈ A.1 Examples include the inside semir- example, the probabilistic CKY algorithm (run on ing, hR≥0, +, ×, 0, 1i, and the Viterbi semiring, sentence w1w2...wn) is written as hR≥0, max, ×, 0, 1i. The former sums the prob- abilities of all proofs of each theorem. The lat- CX,i−1,i = pX→w (1) i ter (used in Eq. 1) calculates the probability of the CX,i,k = max Y,Z∈N;j∈{i+1,...,k−1} most probable proof of each theorem. Two more examples follow. pX→YZ × CY,i,j × CZ,j,k goal = CS,0,n Viterbi proof semiring. We typically need to where N is the nonterminal set and S ∈ N is the recover the steps in the most probable proof in addition to its probability. This is often done us- start symbol. Each CX,i,j variable corresponds to the chart value (probability of the most likely sub- ing backpointers, but can also be accomplished by tree) of an X-constituent spanning the substring representing the most probable proof for each the- orem in its entirety as part of the semiring value wi+1...wj. goal is a special variable of greatest in- terest, though solving for goal correctly may (in (Goodman, 1999). For generality, we define a general, but not in this example) require solving proof as a string that is constructed from strings for all the other values. We will use the term “in- associated with axioms, but the particular form dex” to refer to the subscript values on variables of a proof is problem-dependent. The “Viterbi proof” semiring includes the probability of the (X, i, j on CX,i,j). Where convenient, we will make use of Shieber most probable proof and the proof itself. Letting ∗ et al.’s logic programming view of dynamic pro- L ⊆ Σ be the proof language on some symbol set Σ, this semiring is defined on the set ≥0 × L gramming. In this view, each variable (e.g., CX,i,j R in Eq. 1) corresponds to the value of a “theo- with 0 element h0, i and 1 element h1, i. For two values hu1,U1i and hu2,U2i, the aggregation rem,” the constants in the equations (e.g., pX→YZ operator returns hmax(u , u ),U i. in Eq. 1) correspond to the values of “axioms,” 1 2 argmaxi∈{1,2} ui and the DP defines quantities corresponding to 1When cycles are permitted, i.e., where the value of one weighted “proofs” of the goal theorem (e.g., find- variable depends on itself, infinite sums can be involved. We ing the maximum-valued proof, or aggregating must ensure that these infinite sums are well defined under the semiring. So-called complete semirings satisfy additional proof values). The value of a proof is a combi- conditions to handle infinite sums, but for simplicity we will nation of the values of the axioms it starts with. restrict our attention to DPs that do not involve cycles. Semiring A Aggregation (⊕) Combination (⊗) 0 1 inside R≥0 u1 + u2 u1u2 0 1 Viterbi R≥0 max(u1, u2) u1u2 0 1 Viterbi proof R≥0 × L hmax(u1, u2),Uargmaxi∈{1,2} ui i hu1u2,U1.U2i h0, i h1, i ≤k k-best proof (R≥0 × L) max-k(u1 ∪ u2) max-k(u1 ? u2) ∅ {h1, i} Table 1: Commonly used semirings. An element in the Viterbi proof semiring is denoted hu1,U1i, where u1 is the probability of proof U1. The max-k function returns a sorted list of the top-k proofs from a set. The ? function performs a cross-product on two k-best proof lists (Eq. 2). The combination operator returns hu1u2,U1.U2i, yˆ ∈ Y for a given input x ∈ X: where U1.U2 denotes the string concatenation of 2 QM hm(x,y) U1 and U2. yˆ(x) = argmaxy∈Y m=1 λm (4) k-best proof semiring. The “k-best proof” The second is the summing problem, which semiring computes the values and proof strings of marginalizes the proof probabilities (without nor- the k most-probable proofs for each theorem. The malization): ≤k set is (R≥0 × L) , i.e., sequences (up to length P QM hm(x,y) k) of sorted probability/proof pairs.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Sequence Models
    Sequence Models Noah Smith Computer Science & Engineering University of Washington [email protected] Lecture Outline 1. Markov models 2. Hidden Markov models 3. Viterbi algorithm 4. Other inference algorithms for HMMs 5. Learning algorithms for HMMs Shameless Self-Promotion • Linguistic Structure Prediction (2011) • Links material in this lecture to many related ideas, including some in other LxMLS lectures. • Available in electronic and print form. MARKOV MODELS One View of Text • Sequence of symbols (bytes, letters, characters, morphemes, words, …) – Let Σ denote the set of symbols. • Lots of possible sequences. (Σ* is infinitely large.) • Probability distributions over Σ*? Pop QuiZ • Am I wearing a generative or discriminative hat right now? Pop QuiZ • Generative models tell a • Discriminative models mythical story to focus on tasks (like explain the data. sorting examples). Trivial Distributions over Σ* • Give probability 0 to sequences with length greater than B; uniform over the rest. • Use data: with N examples, give probability N-1 to each observed sequence, 0 to the rest. • What if we want every sequence to get some probability? – Need a probabilistic model family and algorithms for constructing the model from data. A History-Based Model n+1 p(start,w1,w2,...,wn, stop) = γ(wi w1,w2,...,wi 1) | − i=1 • Generate each word from left to right, conditioned on what came before it. Die / Dice one die two dice start one die per history: … … … start I one die per history: … … … history = start start I want one die per history: … … history = start I … start I want a one die per history: … … … history = start I want start I want a flight one die per history: … … … history = start I want a start I want a flight to one die per history: … … … history = start I want a flight start I want a flight to Lisbon one die per history: … … … history = start I want a flight to start I want a flight to Lisbon .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Graphical Models
    Graphical Models Unit 11 Machine Learning University of Vienna Graphical Models Bayesian Networks (directed graph) The Variable Elimination Algorithm Approximate Inference (The Gibbs Sampler ) Markov networks (undirected graph) Markov Random fields (MRF) Hidden markov models (HMMs) The Viterbi Algorithm Kalman Filter Simple graphical model 1 The graphs are the sets of nodes, together with the links between them, which can be either directed or not. If two nodes are not linked, than those two variables are independent. The arrows denote causal relationships between nodes that represent features. The probability of A and B is the same as the probability of A times the probability of B conditioned on A: P(a; b) = P(bja)P(a) Simple graphical model 2 The nodes are separated into: • observed nodes: where we can see their values directly • hidden or latent nodes: whose values we hope to infer, and which may not have clear meanings in all cases. C is conditionally independent of B, given A Example: Exam Panic Directed acyclic graphs (DAG) paired with the conditional probability tables are called Bayesian networks. B - denotes a node stating whether the exam was boring R - whether or not you revised A - whether or not you attended lectures P - whether or not you will panic before the exam Example: Exam Panic R P(r) P(:r) P(b) P(:b) T 0.3 0.7 0.5 0.5 F 0.8 0.2 RA P(p) P(:p) A P(a) P(:a) TT 0 1 T 0.1 0.9 TF 0.8 0.2 F 0.5 0.5 FT 0.6 0.4 FF 1 0 P The probability of panicking: P(p) = b;r;a P(b; r; a; p) = P b;r;a P(b) × P(rjb) × P(ajb) × P(pjr; a) Example: Exam Panic Suppose you know that the course was boring, and want to work out how likely it is that you will panic before exam.
    [Show full text]
  • Hidden Markov Models in Bioinformatics
    Current Bioinformatics, 2007, 2, 49-61 49 Hidden Markov Models in Bioinformatics Valeria De Fonzo1, Filippo Aluffi-Pentini2 and Valerio Parisi*,3 1EuroBioPark, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy 2Dipartimento Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy 3Dipartimento di Medicina Sperimentale e Patologia, Università di Roma “La Sapienza”, Viale Regina Elena 324, 00161 Roma, Italy Abstract: Hidden Markov Models (HMMs) became recently important and popular among bioinformatics researchers, and many software tools are based on them. In this survey, we first consider in some detail the mathematical foundations of HMMs, we describe the most important algorithms, and provide useful comparisons, pointing out advantages and drawbacks. We then consider the major bioinformatics applications, such as alignment, labeling, and profiling of sequences, protein structure prediction, and pattern recognition. We finally provide a critical appraisal of the use and perspectives of HMMs in bioinformatics. Keywords: Hidden markov model, HMM, dynamical programming, labeling, sequence profiling, structure prediction. INTRODUCTION this case, the time evolution of the internal states can be induced only through the sequence of the observed output A Markov process is a particular case of stochastic states. process, where the state at every time belongs to a finite set, the evolution occurs in a discrete time and the probability If the number of internal states is N, the transition distribution of a state at a given time is explicitly dependent probability law is described by a matrix with N times N only on the last states and not on all the others.
    [Show full text]
  • Hidden Markov Model: Tutorial
    Hidden Markov Model: Tutorial Benyamin Ghojogh [email protected] Department of Electrical and Computer Engineering, Machine Learning Laboratory, University of Waterloo, Waterloo, ON, Canada Fakhri Karray [email protected] Department of Electrical and Computer Engineering, Centre for Pattern Analysis and Machine Intelligence, University of Waterloo, Waterloo, ON, Canada Mark Crowley [email protected] Department of Electrical and Computer Engineering, Machine Learning Laboratory, University of Waterloo, Waterloo, ON, Canada Abstract emitted observation symbols (Rabiner, 1989). HMM mod- This is a tutorial paper for Hidden Markov Model els the the probability density of a sequence of observed (HMM). First, we briefly review the background symbols (Roweis, 2003). on Expectation Maximization (EM), Lagrange This paper is a technical tutorial for evaluation, estima- multiplier, factor graph, the sum-product algo- tion, and training in HMM. We also explain some applica- rithm, the max-product algorithm, and belief tions for HMM. There exist many different applications for propagation by the forward-backward procedure. HMM. One of the most famous applications is in speech Then, we introduce probabilistic graphical mod- recognition (Rabiner, 1989; Gales et al., 2008) where an els including Markov random field and Bayesian HMM is trained for every word in the speech (Rabiner & network. Markov property and Discrete Time Juang, 1986). Another application of HMM is in action Markov Chain (DTMC) are also introduced. We, recognition (Yamato et al., 1992) because an action can then, explain likelihood estimation and EM in be seen as a sequence or times series of poses (Ghojogh HMM in technical details. We explain evalua- et al., 2017; Mokari et al., 2018).
    [Show full text]
  • Lecture 6A: Introduction to Hidden Markov Models
    Lecture 6a: Introduction to Hidden Markov Models Introduction to Computational Biology Instructor: Teresa Przytycka, PhD Igor Rogozin PhD First order Markov model (informal) α A G β β β β β,α -probability of given mutation in a unit of time C T α transversion transition A random walk in this graph will generates a path; say AATTCA…. For each such path we can compute the probability of the path In this graph every path is possible (with different probability) but in general this does need to be true. First order Markov model (formal) Markov model is represented by a graph with set of vertices corresponding to the set of states Q and probability of going from state i to state j in a random walk described by matrix a: a – n x n transition probability matrix a(i,j)= P[q t+1=j|q t=i] where q t denotes state at time t Thus Markov model M is described by Q and a M = (Q, a) Example 1/4 Transition probabilities for general DNA seq. Set of states Q: {Begin, End, A,T,C,G} Transition probability matrix a: a Probability of a sequence of states S=q0,…,qm in model M Assume a random walk that starts from state q0 and goes for m steps. What is the probability that S= q0,…,qm is the sequence of states visited? P(S|M) = a(q0,q1) a(q1 , q2) a(q2 , q3) … a(qm-1 , qm) P(S|M) = probability of visiting sequence of states S assuming Markov model M defined by a: n x n transition probability matrix a(i,j) = Pr[q t+1=j|q t=i] Example 1/4 Transition probabilities for general DNA seq.
    [Show full text]
  • Hidden Markov Models + Conditional Random Fields
    10-715 Advanced Intro. to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Hidden Markov Models + Conditional Random Fields Matt Gormley Guest Lecture 2 Oct. 31, 2018 1 1. Data 2. Model (n) N 1 = x n=1 p(x ✓)= (x ) D { } | Z(✓) C C n v p d n Sample 1: C time flies like an arrow Y2C T1 T2 T3 T4 T5 Sample n n v d n 2: W W W W W time flies like an arrow 1 2 3 4 5 Sample n v p n n 3: flies fly with their wings 3. Objective N Sample p n n v v (n) 4: `(✓; )= log p(x ✓) with time you will see D | n=1 X 5. Inference 4. Learning 1. Marginal Inference p(x )= p(x0 ✓) ✓⇤ = argmax `(✓; ) C | x :x =x D 0 C0 C ✓ 2. Partition Function X Z(✓)= C (xC ) x C 3. MAP Inference X Y2C xˆ = argmax p(x ✓) x | 2 HIDDEN MARKOV MODEL (HMM) 3 Dataset for Supervised Part-of-Speech (POS) Tagging Data: = x(n), y(n) N D { }n=1 n v p d n y(1) Sample 1: time flies like an arrow x(1) n n v d n y(2) Sample 2: time flies like an arrow x(2) (3) n v p n n y Sample 3: (3) flies fly with their wings x (4) p n n v v y Sample 4: (4) with time you will see x 4 Naïve Bayes for Time Series Data We could treat each word-tag pair (i.e.
    [Show full text]
  • 4A. Inference in Graphical Models Inference on a Chain (Rep.)
    Computer Vision Group Prof. Daniel Cremers 4a. Inference in Graphical Models Inference on a Chain (Rep.) • The first values of µα and µβ are: • The partition function can be computed at any node: • Overall, we have O(NK2) operations to compute the marginal Machine Learning for Dr. Rudolph Triebel Computer Vision 2 Computer Vision Group More General Graphs The message-passing algorithm can be extended to more general graphs: Directed Undirected Polytree Tree Tree It is then known as the sum-product algorithm. A special case of this is belief propagation. Machine Learning for Dr. Rudolph Triebel Computer Vision 3 Computer Vision Group More General Graphs The message-passing algorithm can be extended to more general graphs: Undirected Tree An undirected tree is defined as a graph that has exactly one path between any two nodes Machine Learning for Dr. Rudolph Triebel Computer Vision 4 Computer Vision Group More General Graphs The message-passing algorithm can be extended to more general graphs: Directed A directed tree has Tree Conversion from only one node a directed to an without parents and undirected tree is all other nodes no problem, have exactly one because no links parent are inserted The same is true for the conversion back to a directed tree Machine Learning for Dr. Rudolph Triebel Computer Vision 5 Computer Vision Group More General Graphs The message-passing algorithm can be extended to more general graphs: Polytree Polytrees can contain nodes with several parents, therefore moralization can remove independence relations Machine Learning for Dr. Rudolph Triebel Computer Vision 6 Computer Vision Group Factor Graphs • The Sum-product algorithm can be used to do inference on undirected and directed graphs.
    [Show full text]
  • Hidden Markov Models
    Hidden Markov Models A profile HMM What is a HMM? • It’s a model. It condenses information. • Models 1D discrete data. • Directed graph. • Nodes emit discrete data • Edges transition between nodes. • What’s hidden about it? • Node identity. • Who is Markov? 2 Markov processes time sequence Markov process is any process where the next item in the list depends on the current item. The dimension can be time, sequence position, etc Modeling protein secondary structure using Markov chains “states” connected by “transitions” H E H=helix E=extended (strand) L L=loop Setting the parameters of a Markov model from data. Secondary structure data LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLEEEEELLLLEEEEELLLLLLLLLLLEEEEEEEEELLLLLEEEEEEEEELL LLLLLLEEEEEELLLLLEEEEEELLLLLLLLLLLLLLLEEEEEELLLLLEEEEELLLLLLLLLLEEEELLLLEEEELLLLEEE EEEEELLLLLLEEEEEEEEELLLLLLEELLLLLLLLLLLLLLLLLLLLLLLLEEEEELLLEEEEEELLLLLLLLLLEEEEEEL LLLLEEELLLLLLLLLLLLLEEEEEEEEELLLEEEEEELLLLLLLLLLLLLLLLLLLHHHHHHHHHHHHHLLLLLLLEEL HHHHHHHHHHLLLLLLHHHHHHHHHHHLLLLLLLELHHHHHHHHHHHHLLLLLHHHHHHHHHHHHHLLLLLEEEL HHHHHHHHHHLLLLLLHHHHHHHHHHEELLLLLLHHHHHHHHHHHLLLLLLLHHHHHHHHHHHHHHHHHHHHH HHHHHHLLLLLLHHHHHHHHHHHHHHHHHHLLLHHHHHHHHHHHHHHLLLLEEEELLLLLLLLLLLLLLLLEEEEL LLLHHHHHHHHHHHHHHHLLLLLLLLEELLLLLHHHHHHHHHHHHHHLLLLLLEEEEELLLLLLLLLLHHHHHHHH HHHHHHHHHHHHHHHLLLLLHHHHHHHHHLLLLHHHHHHHLLHHHHHHHHHHHHHHHHHHHH Count the pairs to get the transition probability. E P(L|E) L P(L|E) = P(EL)/P(E) = counts(EL)/counts(E) counts(E) = counts(EE) + counts(EL) + counts(EH) Therefore: P(E|E) + P(L|E) + P(H|E) = 1. A transition matrix P(H|H)
    [Show full text]