lecture19 - Machine Learning 10-701/15-781, Fall 2008 10-...

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1 Eric Xing © Eric Xing @ CMU, 2006-2008 1 Machine Learning Machine Learning 10 10 -701/15 701/15 -781, Fall 2008 781, Fall 2008 Markov Chain Monte Carlo Markov Chain Monte Carlo And And Topic Models Topic Models Eric Xing Eric Xing Lecture 19, November 17, 2008 Reading: Chap. 8, C.B book Eric Xing © Eric Xing @ CMU, 2006-2008 2 Approaches to inference z Exact inference algorithms z The elimination algorithm z The junction tree algorithms (but will not cover in detail here) z Approximate inference techniques z Stochastic simulation / sampling methods z Markov chain Monte Carlo methods z Variational algorithms (later lectures)
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2 Eric Xing © Eric Xing @ CMU, 2006-2008 3 Monte Carlo methods z Draw random samples from the desired distribution z Yield a stochastic representation of a complex distribution z marginals and other expections can be approximated using sample-based averages z Asymptotically exact and easy to apply to arbitrary models z Challenges: z how to draw samples from a given dist. (not all distributions can be trivially sampled)? z how to make better use of the samples (not all sample are useful, or eqally useful, see an example later)? z how to know we've sampled enough? = = N t t x f N x f E 1 1 ) ( )] ( [ ) ( Eric Xing © Eric Xing @ CMU, 2006-2008 4 Example: naive sampling z Sampling: Construct samples according to probabilities given in a BN. Alarm example: (Choose the right sampling sequence) 1) Sampling:P(B)=<0.001, 0.999> suppose it is false, B0. Same for E0. P(A|B0, E0)=<0.001, 0.999> suppose it is false. ..
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3 Eric Xing © Eric Xing @ CMU, 2006-2008 5 Example: naive sampling z Sampling: Construct samples according to probabilities given in a BN. Alarm example: (Choose the right sampling sequence) 1) Sampling:P(B)=<0.001, 0.999> suppose it is false, B0. Same for E0. P(A|B0, E0)=<0.001, 0.999> suppose it is false. .. 2) Frequency counting: In the samples right, P(J|A0)=P(J,A0)/P(A0)=< 1/9 , 8/9 >. J1 M1 A1 B0 E1 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J1 M0 A0 B0 E0 J0 M0 A0 B0 E0 Eric Xing © Eric Xing @ CMU, 2006-2008 6 Example: naive sampling z Sampling: Construct samples according to probabilities given in a BN. Alarm example: (Choose the right sampling sequence) 3) what if we want to compute P(J|A1) ? we have only one sample . .. P(J|A1)=P(J,A1)/P(A1)=<0, 1>. 4) what if we want to compute P(J|B1) ? No such sample available! P(J|A1)=P(J,B1)/P(B1) can not be defined. For a model with hundreds or more variables, rare events will be very hard to garner evough samples even after a long time or sampling . .. J1 M1 A1 B0 E1 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J0 M0 A0 B0 E0 J1 M0 A0 B0 E0 J0 M0 A0 B0 E0
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4 Eric Xing © Eric Xing @ CMU, 2006-2008 7 Monte Carlo methods (cond.) z Direct Sampling z We have seen it. z Very difficult to populate a high-dimensional state space z Rejection Sampling z Create samples like direct sampling, only count samples which is consistent with given evidences.
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This note was uploaded on 01/26/2010 for the course MACHINE LE 10701 taught by Professor Ericp.xing during the Fall '08 term at Carnegie Mellon.

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lecture19 - Machine Learning 10-701/15-781, Fall 2008 10-...

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