Probabilistic Inference in PRISM: Solving Statistical Machine Learning Challenges
Introducing PRISM by Taisuke Sato from Tokyo Institute of Technology, a high-level modeling language that simplifies the labor-intensive process of statistical machine learning. From basic ideas to ABO blood type program, discover how PRISM offers universal learning and inference methods applicable to every model, revolutionizing probabilistic inference.
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Probabilistic Inference in PRISM Taisuke Sato Tokyo Institute of Technology
Problem Statistical machine learning is a labor-intensive process: {modeling learning evaluation}* of trial-and-error Pains of deriving and implementing model-specific learning algorithms and model-specific probabilistic inference ... Model 1 Model 2 Model n EM2 EM1 EMn ... EM VB MCMC model-specific learning algorithms
Our solution Develop a high-level modeling language that offers universal learning and inference methods applicable to every model ... Model 1 Model 2 Model n modeling language ... EM VB MCMC The user concentrates on modeling and the rest (learning and inference) is taken care of by the system
PRISM (http://sato-www.cs.titech.ac.jp/prism/) Logic-based high-level modeling language Probabilistic models Bayesian network New model HMM PCFG ... PRISM system EM/MAP VT VBVT VB MCMC Learning methods Its generic inference/learning methods subsume standard algorithms such as FB for HMMs and BP for Bayesian networks
Basic ideas Semantics program = Turing machine + probabilistic choice + Dirichlet prior denotation = a probability measure over possible worlds Propositionalized probability computation (PPC) programs written at predicate logic level probability computation at propositional logic level Dynamic programming for PPC proof search generates a directed graph (explanation graph) Probabilities are computed from bottom to top in the graph Discriminative use generatively define a model by a PRISM program and descriminatively use it for better prediction performance
ABO blood type program values(abo,[a,b,o],[0.5,0.2,0.3]). msw(abo,a) is true with prob. 0.5 btype(X):- gtype(Gf,Gm), pg_table(X,[Gf,Gm]). pg_table(X,GT):- ((X=a;X=b),(GT=[X,o];GT=[o,X];GT=[X,X]) ; X=o,GT=[o,o] ; X=ab,(GT=[a,b];GT=[b,a])). gtype(Gf,Gm):- msw(abo,Gf),msw(abo,Gm). father mother a b a o AB A child b o B probabilistic primitives simulate gene inheritance from father (left) and mother (right)
Propositionalized probability computation Explanation graph for btype(a) that explains how btype(a) is proved by probabilistic choice made by msw-atoms 0.55 btype(a) <=> gtype(a,a) v gtype(a,o) v gtype(o,a) 0.15 0.15 0.25 0.25 0.5 0.5 gtype(a,a) <=> msw(abo,a) & msw(abo,a) 0.15 0.5 0.3 gtype(a,o) <=> msw(abo,a) & msw(abo,o) Sum-product computation of probabilities in a bottom-up manner using probabilities assigned to msw atoms 0.15 0.3 0.5 gtype(o,a) <=> msw(abo,o) & msw(abo,a) PPC+DP subsumes forward-backward, belief propagation, inside- outside computation Expl. graph is acyclic and dynamic programming (DP) is possible
Learning A program defines a joint distributionP(x,y| ) where x hidden and y observed P(msw(abo,a),..btype(a), | a, b, o) where a+ b+ o=1 Learning from observed data y by maximizing P(y| ) MLE/MAP P(x*,y| ) where x* = argmax_xP(x,y| ) VT From a Bayesian point of view, a program defines marginal likelihood P(x,y| , ) d We wish to compute predictive distribution = P(x|y, , ) d marginal likelihood P(y| ) = x P(x,y| , ) d Both need approximation Variational Bayes (VB) VB, VB-VT MCMC Metropolis-Hastings
Sample session 1 - Expl. graph and prob. computation built-in predicate | ?- prism(blood) loading::blood.psm.out | ?- show_sw Switch gene: unfixed_p: a (p: 0.500000000) b (p: 0.200000000) o (p: 0.300000000) | ?- probf(btype(a)) btype(a) <=> gtype(a,a) v gtype(a,o) v gtype(o,a) gtype(a,a) <=> msw(gene,a) & msw(gene,a) gtype(a,o) <=> msw(gene,a) & msw(gene,o) gtype(o,a) <=> msw(gene,o) & msw(gene,a) | ?- prob(btype(a),P) P = 0.55
Sample session 2 - MLE and Viterbi inference | ?- D=[btype(a),btype(a),btype(ab),btype(o)],learn(D) Exporting switch information to the EM routine ... done #em-iters: 0(4) (Converged: -4.965121886) Statistics on learning: Graph size: 18 Number of switches: 1 Number of switch instances: 3 Number of iterations: 4 Final log likelihood: -4.965121886 | ?- prob(btype(a),P) P = 0.598211 | ?- viterbif(btype(a)) btype(a) <= gtype(a,a) gtype(a,a) <= msw(gene,a) & msw(gene,a)
Sample session 3 - Bayes inference by MCMC | ?- D=[btype(a), btype(a), btype(ab), btype(o)], marg_mcmc_full(D,[burn_in(1000),end(10000),skip(5)],[VFE,ELM]), marg_exact(D,LogM) VFE = -5.54836 ELM = -5.48608 LogM = -5.48578 |?- D=[btype(a), btype(a), btype(ab) ,btype(o)], predict_mcmc_full(D,[btype(a)],[[_,E,_]]), print_graph(E,[lr('<=')]) btype(a) <= gtype(a,a) gtype(a,a) <= msw(gene,a) & msw(gene,a)
Summary PRISM = Probabilistic Prolog for statistical machine learning Forward sampling Exact probability computation Parameter learning MLE/MAP, VT Bayesian inference VB VBVT MCMC Viterbi inference model core (BIC,Cheesman-Stutz,VFE) smoothing Current version 2.1
