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This chapter discusses various sampling algorithms for approximate inference in Bayesian Networks, including direct sampling, rejection sampling, likelihood sampling, and Markov Chain Monte Carlo (MCMC) simulation. It also explores extending probability to first-order representations and other approaches to uncertain reasoning such as Dempster-Shafer Theory and Fuzzy Logic.
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March 2, 2004 Chapter 14: Probabilistic Reasoning
14.5 Approximate Inference in Bayesian Networks • Monte Carlo Algorithms • randomized sampling • accuracy depends on number of samples generated • e.g., estimating PI
Direct Sampling • Figure 14.2 • P(x1, … xn) = N(x1, … xn) / Total Samples
Rejection Sampling • Used to produce samples from a hard-to-sample distribution, given an easy-to-sample distribution • Figure 14.3 • Drawback: can generate useless samples
Likelihood Sampling • Generates only events that are consistent with the evidence • Figure 14.14 • Page 514
Markov Chain Simulation • MCMC (Markov Chain Monte Carlo) • Make random changes to previous state • Sampling process settles into a dynamic equilibrium in which the long-run fraction of time spent in each state is exactly proportional to its posterior probability
MCMC • Figure 14.15 • Page 516
14.6 Extending Probability to First Order Representations • RPM: Relational Probability Model • Skip!
14.7 Other Approaches to Uncertain Reasoning • Dempster-Shafer Theory • Fuzzy Logic
Dempster-Shafer Theory • Notion of “ignorance” instead of “uncertainty” • reliability of Betty = 90% • reliability of Sally = 80% • Betty and Sally both say a limb fell on my car
Dempster-Shafer Theory • Bel(limb didn’t fall on car) = .1*.2 • Bel(limb fell on car) = 1 - .1*.2 • Betty says its did, Sally says it didn’t • only Betty reliable .9 * .2 = .18 • only Sally reliable .1 * .8 = .08 • neither reliable .1 * .2 = .02 • Bel(limb fell on car) = 18/28 • Bel(limb didn’t fall on car) = 8/28
Fuzzy Logic • Degree of membership • Typical membership function for concept of “warm” • Union • Intersection • Negation
Fuzzy Logic Applications • Air conditioning • Cruise control • Ship boilers • Video cameras • Washing Machines
Fuzzy System Construction • Fuzzificiation • Inference • Composition • Defuzzification