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MCMC based machine learning a . (Bayesian Model Averaging) Nicos Angelopoulos n.angelopoulos@ed.ac.uk School of Biological Sciences Biochemistry Group University of Edinburgh, Scotland, UK. a Collaborative work with James Cussens, York

  1. MCMC based machine learning a . (Bayesian Model Averaging) Nicos Angelopoulos n.angelopoulos@ed.ac.uk School of Biological Sciences Biochemistry Group University of Edinburgh, Scotland, UK. a Collaborative work with James Cussens, York University, jc@cs.york.ac.uk P .E.L. 2006 – p.1

  2. MCMC Overview Class of sampling algorithms that estimate a posterior distribution. Markov chain construct a chain of visited values, M 1 , M 2 , . . . , M n , by proposing M ∗ from M i , with probability q ( M ∗ , M i ) . Use prior knowledge, p ( M ∗ ) and relative likelihood of the two values, p ( D | M ∗ ) /p ( D | M i ) to decide chain construction. Monte Carlo Use the chain to approximate the posterior p ( M | D ) . P .E.L. 2006 – p.2

  3. Bayesian learning with MCMC Given some data D and a class of statistical models M ( M ∈ M ) that can express relations in the data, use MCMC to approximate normalisation factor in Bayes’ theorem p ( D | M ) p ( M ) p ( M | D ) = � M p ( D | M ) p ( M ) p ( M ) is the prior probability of each model p ( D | M ) the likelihood (how well the model fits the data) p ( M | D ) the posterior P .E.L. 2006 – p.3

  4. Example: Data smoker bronchitis l_cancer person 1 y y n person 2 y n n person 3 y y y person 4 n y n person 5 n n n P .E.L. 2006 – p.4

  5. Example: Models S B 1 B L [b-[],l-[],s-[]] S B 2 B L [b-[s],l-[],s-[]] . . . S B 24 [b-[s],l-[b,s],s-[]] B L P .E.L. 2006 – p.5

  6. Example: Objective P(Bx) . . . B1 B2 B3 B4 . . . B24 � p ( B x ) = 1 B x P .E.L. 2006 – p.6

  7. Metropolis-Hastings (M-H) MCMC 0. Set i = 0 and find M 0 using the prior. 1. From M i produce a candidate model M ∗ . Let the probability of reaching M ∗ be q ( M ∗ , M i ) . 2. Let � q ( M ∗ , M i ) P ( D | M ∗ ) P ( M ∗ ) � α ( M i , M ∗ ) = min q ( M i , M ∗ ) P ( D | M i ) P ( M i ) , 1 � M ∗ with probability α ( M i , M ∗ ) M i +1 = M i with probability 1 − α ( M i , M ∗ ) 3. If i reached limit then terminate, else set i = i + 1 and repeat from 1. P .E.L. 2006 – p.7

  8. Example: MCMC Markov Chain: M 1 B 3 P .E.L. 2006 – p.8

  9. Example: MCMC Markov Chain: M 1 , M 2 B 3 , B 3 P .E.L. 2006 – p.8

  10. Example: MCMC Markov Chain: M 1 , M 2 , M 3 , M 4 , M 5 , . . . B 3 , B 3 , B 10 , B 3 , B 24 , . . . P .E.L. 2006 – p.8

  11. Example: MCMC Markov Chain: M 1 , M 2 , M 3 , M 4 , M 5 , . . . B 3 , B 3 , B 10 , B 3 , B 24 , . . . Monte Carlo: #( B k ) p ( B k ) = � B x #( B x ) P .E.L. 2006 – p.8

  12. SLP defined model space ?− bn( [1,2,3], Bn ). G0 Gi Mi M* From M i identify G i then sample forward to M ⋆ . q ( M i , M ⋆ ) is the probability of proposing M ⋆ when M i is the current model. P .E.L. 2006 – p.9

  13. BN Prior bn( OrdNodes, Bn ) :- bn( Nodes, [], Bn ). bn( [], _PotPar, [] ). bn( [H|T], PotPar, [H-SelParOfH|RemBn] ) :- select_parents( PotPar, H, SelParOfH ), bn( T, [H|PotPar], RemBn ). select_parents( [], [] ). select_parents( [H|T], Pa ) :- include_element( H, Pa, RemPa ), select_parents( T, TPa ). 1/2 : include_element( H, [H|TPa], TPa ). 1/2 : include_element( _H, TPa, TPa ). P .E.L. 2006 – p.10

  14. example BN (Asia) For example ? - bn( [1,2,3,4,5,6,7,8], M ). M = [1-[],2-[1],3-[2,5],4-[],5-[4],6-[4],7-[3],8-[3,6]]. P .E.L. 2006 – p.11

  15. visits and stays P .E.L. 2006 – p.12

  16. Edges recovery With topological ordering constraint and a maximum of 2 parents per node, the algorithm recovers most of the BN arcs in 0.5 M iterations. For example for a .99 cut -off we have : Missing : 2 → 3 (.84) 3 → 7 (.47) Superfluous : 5 → 7 P .E.L. 2006 – p.13

  17. CART priors ? - cart( M ). x2 P split ( η ) = α (1 + d η ) − β =< 1 1 < x1 =< 0 0 < M = node( b, 1, node(a,0,leaf,leaf), leaf ) 1 - Sp: [Sp]: cart( Data, D, A/B, leaf(Data) ). Sp: [Sp]: cart( Data, D, A/B, node(F,V,L,R) ) :- branch( Data, F, V, LData, RData ), D1 is D + 1, NxtSp is A * ((1 + D1) ˆ -B), [NxtSp] : cart( LData, D1, A/B, L ), [NxtSp] : cart( RData, D1, A/B, R ). P .E.L. 2006 – p.14

  18. Experiment Pima Indians Diabetes Database 768 complete entries of 8 variables. Denison et.al. run 250,000 iterations of local perturbations. Their best likelihood model: -343.056 Our experiment run for 250,000 iterations with branch replacing. Parameters: uniform choice proposal, α = . 95 β = . 8 Our best likelihood model: -347.651 P .E.L. 2006 – p.15

  19. Likelihoods trace -340 ’tr_uc_rm_pima_idsd_a0_95b0_8_i250K__s776.llhoods’ -350 -360 -370 -380 -390 -400 -410 -420 0 50000 100000 150000 200000 250000 β = . 8 , α = . 95 , proposal = uniform choice P .E.L. 2006 – p.16

  20. -347.61529077520584 Best likelihood 1:6 best_llhood:vst(37):msclf(145) =< 29.3 >29.3 2:8 11:2 =< 27 >27 =< 166 >166 3:141/5 4:6 12:8 31:7/62 =< 26.3 >26.3 =< 29 >29 5:2 8:8 13:2 28:2 =< 152 >152 =< 54 >54 =< 127 >127 =< 106 >106 6:54/3 7:2/9 9:22/24 10:10/1 14:6 17:3 29:52/19 30:53/92 =< 45.4 >45.4 =< 61 >61 15:129/19 16:1/4 18:1/11 19:8 =< 27 >27 20:5 27:7/5 =< 200 >200 21:7 26:12/1 =< 0.314 >0.314 22:4/1 23:5 =< 32 >32 24:4/6 25:1/6 P .E.L. 2006 – p.17

  21. in Kyoto Models: HMRFs for clustering. Likelihood: design and implement a likelihood -ratio function for HMRFs. Proposal: implement function(s) for reaching proposal model. Application: to real data. SLPs: for more complex priors. P .E.L. 2006 – p.18

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