Inference in Bayesian network • Exact inference algorithms: – Variable elimination – Recursive decomposition (Cooper, Darwiche) – Belief propagation algorithm (Pearl) – Arc reversal (Olmsted, Schachter) • Approximate inference algorithms: – Monte Carlo methods: • Forward sampling, Likelihood sampling – Variational methods Crucially, whether inference is tractable depends on the structure of the graph that describes that probability. Question: Are X and Y conditionally independent given evidence vars {Z}? exact and approximate inference in continu-ous time Bayesian networks is NP-hard even when the initial states are given. The GP Model¶. Message Passing & Belief propagation, Hidden Markov Models & Forward-Backward algorithm. This is [2018]. VI comes without warranty. of passengers in "000 600 ML-II vs HMC 1958 1960 Years No. Bayes Ne Representation 3 . The latter one is not used in the usual exact inference. It is used, for example, in expectation propagation or variational Bayes, where inferring an exact posterior is intractable and additional assumptions on the posterior have to be made. In this case, the inferred posterior is approximate. The exact variational principle is intractable to solve. Outline Artificial Intelligence Knowledge Reasoning Machine … In this case, approximate inference using penalized quasi-likelihood approach of Breslow and Clayton [4] or a fully Bayesian approach [5] is usually adopted. Reliable Exact Inference Maximum-a-posteriori (MAP) estimates No Clear Winner Both use unreliable approximate inference approaches Tractability helps achieve better marginal predictions even though the model t is inferior. 2 Propagation w. Approximate Msgs General idea Perform BP (or GBP) as before, but propagate messages that are only approximate Modular approach General inference scheme remains the same Can plug in many different approximate message computations 3 Factorized Messages X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 Markov network 21 X 11 X 12 X 31 X 32 22 X 31 X 11 X 13 (MCMC) methods for performing approximate inference. “Exact” vs. “Approximate” Inference Solver Most theory works for “Exact” Inference Theory breaks with “Approximate” Inference Alex Kulesza, Fernando Pereira: Structured Learning with Approximate Inference. Approximate inference is key to modern probabilistic modelling, and since the start of my PhD there has been considerable progress on this subject. MCMC is asymptotically exact; VI is not. –An overview of what we have seen before –Combinatorial optimization –Different views of inference • Graph algorithms –Dynamic programming, greedy algorithms, search • Integer programming • Heuristics for inference –Sampling • Learning to search 8 Exact Inference: Variable Elimination ! 4 . M ≈( ) ... exact … The latter one is not used in the usual exact inference. Approximate decoding with local search. We construct an approximation Qto the target distribution P. The approximation Qcan be obtained by 1 Selecting a simpler form for Qthat can be e ciently tuned to P. Bayesian inference is a pretty classical problem in statistics and machine learning that relies on the well known Bayes theorem and whose main drawback lies, most of the time, in some very heavy computations. Third, we desire approximations that do not apriori sacri˝ce covariance structure in the pa-127 rameter posteriors, a limitation often induced for tractability in variational approaches to 128 approximate inference (Blei et … Approximate Inference For all but the simplest statistical models, exact learning and inference are computationally intractable. Use approximate inference, trading o computational cost vs. accuracy. Approximate Inference in Fully Bayesian GPR 1958 1960 Years 400 No. But it’s still a variable, just like all theXis, so the distinction is not important now. Approximate Bayesian Inference seems to be used ambiguously, either as a synonym for ABC, or for Variational Inference, i.e. In this thesis we explore both exact inference and iterative approximate inference ap-proaches using adaptive updates. Q&A for people interested in statistics, machine learning, data analysis, data mining, and data visualization uations where the model is being used iteratively; for example in approximate inference we may want to decompose the problem into simpler inference subproblems that are solved repeatedly and iteratively using adaptive updates. Approximate inference. A (\theta) A(θ) In summary: Variational methods in general turn inference into an optimization problem via exponential families and convex duality. Yes, if X and Y “separated” by Z ! (1) In a prediction setting, the goal of approximate inference is to compute efficiently a prediction with the highest possible score. Complexity of Exact Inference Singly connected networks (or polytrees) Any two nodes are connected by at most one path Time and space cost of variable elimination are O(dkn) Multiply connected networks Can reduc 3SAT to exact inference =⇒ NP-hard Equivalent to counting 3SAT models =⇒ #P-complete And apply it to text-mining algorithm called Latent Dirichlet Allocation. In particular, the main novel technical contributions of this thesis are as follows: a way of representing Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3 ), where In practice however, exact inference is intractable for even moderately sized CSPs and approximate inference techniques are essential for obtaining estimates of marginal probabilities. References for the exact posterior Need good reference to assess inference. There are two types of inference techniques: exact inference and approximate inference. Loopy Belief Propagation. Exact inference is carried out with the posterior probability of the parameters. of passengers in "000 ML-II vs FR 1958 1960 Years No. 19 (UC3M)Approximate Inference May 21, 2018 11 / 32. Due to the inference being approximate, the model does not optimize the exact likelihood; instead it optimizes a lower bound of the data log-likelihood. Exact inference algorithms calculate the exact value of probability P(XjY). Thefirst key point we need t… Simply because for many cases, we cannot directly compute the posterior distribution, i.e. the posterior is on an intractable form — often involving integrals — which cannot be (easily) computed. This post focuses on the simplest approach to Variational Inference based on mean-field approximation. Structure learning. In this regard, one of the well-known and accepted methods of performing approximate Bayesian inference is variational inference (VI) ( graves2011practical ; hinton1993keeping ) in which an approximating distribution is used instead of the true Bayesian posterior over model … Exact statistics, such as that described in exact test, is a branch of statistics that was developed to provide more accurate results pertaining to statistical testing and interval estimation by eliminating procedures based on asymptotic and approximate statistical methods. Xiang Zhai Department of Computer Science University of Illinois, Urbana-Champaign 1 Approximate best predictions for the random effects, along with predic- tion variances, also arise quite simply from the Gaussian approximation. The exact variational principle is intractable to solve. As we will see, Inference has a close connection to. use of models and (incomplete) data so to infer knowledge about the state of the world. Approximate sampling with MCMC. Approximate inference is key to modern probabilistic modeling. In general, VI is faster. Abstract Over the past few years, a number of approximate inference algorithms for networked data have been put forth. 14.5 Approximate Inference • Given the intractability of exact inference, it is essential to consider approximate inference methods • Approximation is based on random sampling from a known probability distribution (Monte Carlo algorithms) • E.g., an unbiased coin can be thought of as a random variable In this paper, we present algorithms for adaptive exact inference on general graphs that can be used to efficiently compute marginals and update MAP configurations under arbitrary changes to the input factor graph and its associated elimination tree. of passengers in "000 ML-II vs FR 1958 1960 Years No. In this thesis we explore both exact inference and iterative approximate inference approaches using adaptive updates. Meaning, when we have computational time to kill … • Exact inference by enumeration • Exact inference by variable elimination • Approximate inference by stochastic simulation • Approximate inference by Markov chain Monte Carlo (MCMC) Two random variables A B are (absolutely) independent iff P(ÄIB) = P(Ä) e.g., A and B are two coin tosses This week we will move on to approximate inference methods. ual.es. Hobert [3]). Given a scoring model S(y|x) over candidate labelings y for input x, exact Viterbi inference is the computation of the optimal labeling h(x) = argmax y S(y|x) . Thanks to the availability of big data, significant computational power, and sophisticated models, machine learning has achieved many breakthroughs in multiple application domains. Background. Bayesian Networks Structured, graphical representation of probabilistic relationships between several random variables . PART II - Inference. In the CO Approximate Inference •Speeding Up Inference –Pull out terms –Maximize Independence –Variable Enumeration •Approximate Inference by –Sampling –Rejection sampling –Gibbs sampling. MCMC is computationally expensive. This chapter covers the first exact inference algorithm, variable elimination. Inference • What is inference? Bayesian Inference for Mixture Language Models Chase Geigle, Cheng. Use approximate inference for queries of P θ – Decouples inference from Learning Parameters • Inference is a black-box – But approximation may interfere with learning • Non-convergence of inference can lead to oscillating estimates … This paper. Fast factorisation of probabilistic potentials and its application to approximate inference in Bayesian networks. In general, producing an exact estimate of a conditional probability for a complex probabilistic program is not computationally feasible. Approximate inference methods make it possible to learn realistic models from large data sets. 5 Reachability (D-Separation)! This week we will move on to approximate inference methods. We will also see mean-field approximation in details. OSTI.GOV Journal Article: Approximate inference on planar graphs using loop calculus and belief progagation Title: Approximate inference on planar graphs using loop calculus and belief progagation Full Record Look for active paths from X to Y ! The main idea is that instead of exactly computing the conditional probability distribution (or “posterior”) of interest, we can approximate it … A short summary of … The generalisation of this relation given by our approach makes it possible to apply out of the box infer-ence methods to obtain approximate optimal policies. Maximum Entropy RL - may be it will help to improve exploration! Week 3: Approximate inference withvariationalmethods. Bayesian networks (Bayes nets) • Specify a full joint probability distribution. ing graphical presentation, estimation, and inference in RD designs using local poly-nomial techniques, see Calonico,Cattaneo,andTitiunik (2014a, 2015b). Here’s an example of a Bayesian Network, i.e., a directed graphical model with sevenvariables. Exact Inference by Enumeration Exact Inference by Variable Elimination Approximate Inference by Stochastic Simulation Approximate Inference by Markov Chain Monte Carlo (MCMC) Digging Deeper... Amarda Shehu (580) Outline of Today’s Class { Bayesian Networks and Inference 2. Exact Inference vs. However, performing exact Bayesian inference in deep neural networks is computationally intractable. MCMC is asymptotically exact; VI is not. Approximate Inference IPAM Summer School Ruslan Salakhutdinov BCS, MIT Deprtment of Statistics, University of Toronto 1 The justi cation for conditioning at inference time on margins that were not naturally xed at data sampling time has a long history. involves inference, or reasoning about the state of the underlying variables and quantify-ing the models’ uncertainty about any assignment to them. Exact inference impossible. In the limit, MCMC will exactly approximate the target distribution. We will see why we care about approximating distributions and see variational inference — one of the most powerful methods for this task. Sometimes, exact inference is computationally intractable. Introduction to inference, Exact inference vs approximate inference, Map inference. In the CO approximate inference using a variational distribution. Meaning, when we have computational time to kill … We discuss approximate inference in later chapters. Figure 1. 3 hidden-layer, width 50 BNN vs. GP. Undirected GMs vs. Suppose we want to compute the exact quantity of1 P(X2=x2∣X4=x2). Note that, though the computation of h may be exact,inference is approximate: Slides about I. Goodfellow, Y. Bengio, Deep Learning, Ch. 125 inference scenarios without further cost. Exact inference Approximate inference Learning Bayesian Networks Learning parameters Learning graph structure (model selection) Summary. •First, let us look at some specific examples: – Bayesian Probabilistic Matrix Factorization – Bayesian Neural Networks – Dirichlet Process Mixtures (last class) 8. Approaches to inference 2 •Exact inference algorithms •The elimination algorithm •Message-passing algorithm (sum-product, belief propagation) •Junction tree algorithm •Approximate inference techniques •Variationalalgorithms •Loopy belief propagation •Mean field approximation •Stochastic simulation / sampling methods Exact inference in general graphs is hard •Running time of JT is exponential in the max clique size of the JT Bayesian Networks TU Darmstadt Einführung in die Künstliche Intelligenz 1 V2.0 | J. Fürnkranz Bayesian Networks n Syntax n Semantics n Parametrized Distributions n Inference in Bayesian Networks n Exact Inference nenumeration nvariable elimination n Approximate Inference nstochastic simulation nMarkov Chain Monte Carlo (MCMC) Many slides based on Complexity of Exact Inference 62 Belief (Bayesian) Networks • Motivation • Conditional Independence • Syntax and Semantics • Reasoning with Belief Networks • Construction of Belief Networks • Inference Algorithms • Exact Inferences • Approximate Inference • By Stochastic Simulation • … class of methods that given (trackable) likelihood and priors approximate the posterior distribution. 11 Exact Inference: Elimination Algorithm 12 Exact Inference on Trees: Sum-Product (BP), Sum-product vs Max-product in Factor Graphs 13-14 Examples in Dynamic Models: HMMs and Kalman Filtering 15-16 Exact Inference: Junction Tree Algorithm 17-18 Approximate Inference: Loopy BP 19-20 Approximate Inference: Sampling Methods (MonteCarlo, 1. Request PDF | On Jan 1, 2010, Christopher Russell and others published Exact and Approximate Inference in Associative Hierarchical Networks using Graph Cuts. Variable Elimination and Its complexity, elimination ordering, Junction-Tree Algorithm. Most of what is known about the complexity of CTBNs is derived from the fact that a Bayesian network is used to specify the This comes from the spring 2011 final for Berkeley’s undergraduate AIcourse, and the reasonwhy there’s one Y but six Xis is because the question using this graphical model wanted toemphasize how the Y represented a class variable. What to do then? The key idea MCMC is computationally expensive. Q&A for people interested in statistics, machine learning, data analysis, data mining, and data visualization of passengers in "000 600 ML-II vs MF Figure 1: Time-series (test) predictions under Fully Bayesian GPR vs. ML-II (top: CO 2 and bottom: Airline). Approximate Inference: Belief Propagation • Popular Approach for Approximate Inference is Belief Propagation and its variants – An algorithm from this family would be used for inference in the model resulting from the learning procedure • We should use the same inference … We will see why we care about approximating distributions and see variational inference — one of the most powerful methods for this task. A (\theta) A(θ) In summary: Variational methods in general turn inference into an optimization problem via exponential families and convex duality. There are two distinct components for approximation: Inner or outer bound for the marginal polytope. The ApproximateGP model is GPyTorch’s simplest approximate inference model. 2 Propagation w. Approximate Msgs General idea Perform BP (or GBP) as before, but propagate messages that are only approximate Modular approach General inference scheme remains the same Can plug in many different approximate message computations 3 Factorized Messages X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 Markov network 21 X 11 X 12 X 31 X 32 22 X 31 X 11 X 13 The following is a list of some of the exact and approximate algorithms on graphical models. 1 INTRODUCTION Currently, little theoretical work has been done on the complexity of inference in CTBNs. Variational Inference(VI) is an approximate inference method in Bayesian statistics. We will also see mean-field approximation in details. Variational inference is at the heart of many such approximate inference techniques. In the limit, MCMC will exactly approximate the target distribution. Week 7.

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