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variational bayes in biophysics

This document discusses hierarchical variational Bayesian methods and their applications in modeling biological data, particularly in network analysis. It addresses challenges such as overfitting and resolution limits in community detection within biological and social networks. The presentation is supported by algorithms and formulations for generative models, highlighting the importance of variational methods in inference and model selection.

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bio+stats vbem/networks hierarchical variational and hierarchical modeling for biological data chris wiggins columbia april 23, 2012 chris.wiggins@columbia.edu 4/23/12 Chris Wiggins • APAM: Department of Applied Physics and Applied Mathematics; • C2B2: Center for Computational Biology and Bioinformatics; • CISB: Columbia University Initiative in Systems Biology • ISDE: Institute for Data Sciences and Engineering Columbia University September 28, 2012
bio+stats vbem/networks hierarchical biological challenges inference model selection thanks. . . - jake hofman (vbmod,vbfret) - jonathan bronson (vbfret) - jan-willem van de meent (hfret) - ruben gonzalez (vbfret, hfret) for more info: - vbfret.sourceforge.net - vbmod.sourceforge.net - hfret.sourceforge.net (soon) chris.wiggins@columbia.edu 4/23/12 BMC bioinformatics, 2010; PNAS 2009; Biophysical Journal 2009;
bio+stats vbem/networks hierarchical 1 biology and statistics genomics generative modeling 2 variational/biological networks variational Bayesian expectation maximization inference model selection 3 hierarchical/time series biological challenges inference model selection chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical genomics generative modeling biology and statistics: chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical genomics generative modeling genomics: chris.wiggins@columbia.edu 4/23/12 plos comp bio `10, nyas `07, bmc bioinfo `06a, bmc bioinfo `06b, bioinfo `04, regulatory genomics 04, IEEE `05; NIPS (MLCB`03, `06 )
bio+stats vbem/networks hierarchical genomics generative modeling generative modeling: chris.wiggins@columbia.edu 4/23/12 bmc bioinfo '10, PNAS`09, biophys j`09, PNAS`07, PNAS`06 NIPS (MLCB '09, '10, '11) IEEE sig. proc. `12; plos one `08,cell `07,prl `06, jcs `06, biophys j `06 pami `08, prl `08, NIPS (MLCB08), PNAS `05, bmc bioinfo `04
bio+stats vbem/networks hierarchical vbem inference model selection variational/biological networks: - variational bayesian expectation maximization - inference - model selection chris.wiggins@columbia.edu 4/23/12
introduction formulation results extensions motivation history introduction: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Hartwell, Hopfield, Leibler and Murray NATURE|VOL 402 | SUPP | 2 DECEMBER 1999 | www.nature.com
introduction formulation results extensions motivation history motivation: community detection in networks social networks biological networks problem: over-fitting/resolution limit chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions motivation history history: by community math/cs: spectral methods (Fiedler ’74, Shi + Malik ’00) math/cs: clustering generally (Taskar, Koller, Getoor) physics: modularity common thread: test w/ stochastic block model (’76, ’83) ergo: use as inference tool (Hastings 0604429, Newman+Liecht 061148) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ
introduction formulation results extensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
introduction formulation results extensions generative model max likelihood max evidence algo max likelihood: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model:
introduction formulation results extensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Increasing complexity
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net http://research.microsoft.com/~minka/statlearn/demo/
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net cf. “BIC” Schwartz, 1978
introduction formulation results extensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ
introduction formulation results extensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n
introduction formulation results extensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
introduction formulation results extensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
introduction formulation results extensions generative model max likelihood max evidence algo max likelihood: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥)
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2004 & 2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥) Can do integrals, but sum is intractable, O(KN); use mean-field
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Gibbs’/Jensen’s inequality (log of expected value bounds expected value of log) for any distribution q p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥) Variational Bayes (MacKay, Jordan, Ghahramani, Jaakola, Saul 1999; cf. Feynman 1972)
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A1): Beal, 2003
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A2): Beal, 2003
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A3): Beal, 2003
introduction formulation results extensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Gibbs’/Jensen’s inequality (log of expected value bounds expected value of log) for any distribution q Variational Bayes (MacKay, Jordan, Ghahramani, Jaakola, Saul 1999; cf. Feynman 1972) • F is a functional of q; find approximation to posterior by optimizing approximation to evidence • Take q(z, π, θ)=q(z)q(π)q(θ); Qiμ is probability node i in module μ where expected counts are:
introduction formulation results extensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net where expected counts are:
introduction formulation results extensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net where expected counts are:
introduction formulation results extensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net suggests hard limit in step 3; sparse in step 1
introduction formulation results extensions run time consistency good vs easy real data results: run time consistency required plot: good vs. easy real data karate biology american football chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions run time consistency good vs easy real data run time: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Main loop runtime for 104 nodes in MATLAB ~30 seconds
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 θ N=8, K=2, distribution after 2 iterations p(θ+ ) p(θ − )
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • K=4? • Automatic complexity control: probability of occupation for extraneous modules goes to zero
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • K=4? • Automatic complexity control: probability of occupation for extraneous modules goes to zero
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net The “resolution limit” problem 1 2 3 4 5 57 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 58 59 60 1 2 3 4 5 57 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 58 59 60 Variational Bayes Girvan-Newman modularity
introduction formulation results extensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net The “resolution limit” problem 10 12 14 16 18 20 8 10 12 14 16 18 20 Ktrue K* K * =Ktrue Variational Bayes Modularity optimization 10 12 14 16 18 20 0.72 0.74 0.76 0.78 0.8 0.82 0.84 Ktrue GNmodularity Resolution limit problem on ring of 4−node cliques Single−clique communities (correct) Double−clique communities (incorrect) GN modularity (Clauset’s algorithm) Girvan-Newman modularity or Potts model w/ fixed parameters suffers from a resolution limit, where size of detected modules depends on network size Fortunato et. al. (2007), Kumpula et. al. (2007),
introduction formulation results extensions run time consistency good vs easy real data good vs easy: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions run time consistency good vs easy real data real data: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Correctly infer K=12 conferences Validation: NCAA football schedule 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 nodes: teams edges: games shape: conference color: inferred module
introduction formulation results extensions run time consistency good vs easy real data real data: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net APS march meeting 2008 superconductivity (experimentalists) Nanotubes, Graphene superconductivity (theorists)
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Nodes belong to “blocks” of varying size • Roll die for assignment of nodes to blocks • Probability of edge between two nodes depends only on block membership • Flip (one of K2) coins for edges • Result: mixture of Erdos-Renyi graphs 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2275 adjacency matrix
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net vs
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2803 adjacency matrix
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net vs
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0) running vbmod ... Elapsed time is 1.136925 seconds. running vbsbm ... Elapsed time is 1.398904 seconds. Fmod=13089.158019 Fsbm=13144.445782 vbmod wins
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0.25) running vbmod ... Elapsed time is 1.557298 seconds. running vbsbm ... Elapsed time is 1.759527 seconds. Fmod=20457.142416 Fsbm=19457.306022 vbsbm wins
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0.5) running vbmod ... Elapsed time is 2.624886 seconds. running vbsbm ... Elapsed time is 1.440242 seconds. Fmod=26133.351210 Fsbm=23921.797625 vbsbm wins
introduction formulation results extensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Using same framework we can compare the unconstrained and full stochastic block models via p(D|M,K*) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 perturbation to constrained model winpercentageforunconstrainedmodel 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 perturbation to constrained model winpercentageforunconstrainedmodel 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2100 adjacency matrix 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2048 adjacency matrix 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2108 adjacency matrix
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n L
introduction formulation results extensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
introduction formulation results extensions for more info. . . code: MATLAB & python (inc. “full” SBM) (vbmod.sf.net) paper: arxiv 08 / prl 08 Hofman soon to come (not by me) code in C++, inc. full ‘vblabel propagation’ algo twitter-scale analysis chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
bio+stats vbem/networks hierarchical biological challenges inference model selection hierarchical/time series: - biological challenges - inference - model selection chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Jan-Willem van de Meent, Ruben Gonzalez, Chris Wiggins Columbia University
Ramakrishnan et al – http://www.mrc-lmb.cam.ac.uk/ribo/
Ramakrishnan et al – http://www.mrc-lmb.cam.ac.uk/ribo/
Ramakrishnan et al – http://www.mrc-lmb.cam.ac.uk/ribo/
Ramakrishnan et al – http://www.mrc-lmb.cam.ac.uk/ribo/
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET = cy5 / (cy3 + cy5) Tinoco and Gonzalez, Genes Dev, 2011
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Tinoco and Gonzalez, Genes Dev, 2011
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009 (short-lived GS1 states correspond to an EF-G + GDPNP binding event)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 1. Identify states 2. Estimate Kinetic Rates 3. Average over many time series 4. Detect subpopulations
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET Signal
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram Idea: Find probability of belonging to each state
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Expectation Maximization 1. calculate p(z | x, θi) 2. calculate θi+1 from p(z | x, θi)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Expectation Maximization 1. calculate p(z | x, θi) 2. calculate θi+1 from p(z | x, θi)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Learned Truth Accurate for occupancy of states, not so good for rate estimates
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(x, z µ, σ, π) = p(x z, µ, σ)p(z π)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 probability of state depends on previous state p(zt+ =l zt =k) = Akl
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(z =k) = πk
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(zt+ =l zt =k) = Akl
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(xt zt = k) = N(xt µk , σk)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Learned Real We’ve learned: parameters: θ = {µ, σ, π, A} states: p(z | x, θ)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2 States 3 States
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Log-Evidence
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Prior
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Ensemble
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence best model has highest average likelihood L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Log-Evidence
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Evidence 31 L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Lower Bound L = z ∫ dθ q(z)q(θ w)log p(x, z, θ u) q(z)q(θ w) ≥ log p(x u) q(z)q(θ w) p(z, θ x)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Lower bound tight for true posterior L = z ∫ dθ p(z, θ x)log p(x, z, θ u) p(z, θ x) = z ∫ dθ p(z, θ x)log[p(x u)] = log p(x u) L = log p(x u) − Dkl [q(z)q(θ w) p(z, θ x)]
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 We’ve learned: parameters: q(θ | w) states: p(z | x, θ) δLn δq(zn) = VBEM Updates δLn δq(θn) =
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 variability: photophysical/experimental
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Hierarchical Updates ∂ ∂u n Ln =
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Hierarchical Updates ∂ ∂u n Ln = “two-stage PEB model/CIHM”-Kass Steffey JASA 1989
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2. Update p(θ | u) Until Σ Ln converges Until Ln converges • Update q(zn) • Update q(θn | wn) 1. Run VBEM on each trace δLn δq(zn) = Hierarchical UpdatesVBEM Updates δLn δq(θn) = ∂ ∂u n Ln =
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2. Update p(θ | u) Until Σ Ln converges Until Ln converges • Update q(zn) • Update q(θn | wn) 1. Run VBEM on each trace We’ve learned: p(θn, zn | xn) ≃ q(θn) q(zn) (for each trace) p(θ | u) (for ensemble)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 ξntkl = p(znt = k, znt+ = l xn) 1. Run mixture model on posterior counts p(ξnA) = tkl Aξntkl kl p(ξn um) = ∫ dA p(Aum)p(ξn A) 2. Rerun with M x K block-diagonal form uA = uA uA uA M
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 τfast τslow 2 4 8 16 32 64
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 τfast τslow 2 4 8 16 32 64 reality
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G Fei, Bronson, Hofman, Srinivas, Wiggins, Gonzalez, PNAS, 2009
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G p(zk) ∼ e−Gk kB T log p(zk) − log p(zl ) = −(Gk − Gl )kBT + cst.
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G p(zk) ∼ e−Gk kB T Δ∆G=logit(p)
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-Gbound fraction and life-times
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 5 nM EF-Gbound fraction and life-times
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 50 nM EF-Gbound fraction and life-times
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 250 nM EF-Gbound fraction and life-times
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 500 nM EF-Gbound fraction and life-times
bio+stats vbem/networks hierarchical biological challenges inference model selection model selection: chris.wiggins@columbia.edu 4/23/12
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, UnderfittedInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, CorrectOut vs In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, OverfittedInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, UnderfittedInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, CorrectInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, OverfittedInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, OverfittedInf Out - Inf In
bio+stats vbem/networks hierarchical biological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 the future, in progress: X
bio+stats vbem/networks hierarchical biological challenges inference model selection thanks. . . - jake hofman (vbmod,vbfret) - jonathan bronson (vbfret) - jan-willem van de meent (hfret) - ruben gonzalez (vbfret, hfret) for more info: - vbfret.sourceforge.net - vbmod.sourceforge.net - hfret.sourceforge.net (soon) chris.wiggins@columbia.edu 4/23/12 BMC bioinformatics, 2010; PNAS 2009; Biophysical Journal 2009;
traditional role of statistics in biophysics “if your experiment needs statistics, you ought to have done a better experiment” -lord rutherford

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variational bayes in biophysics

  • 1.
    bio+stats vbem/networks hierarchical variationaland hierarchical modeling for biological data chris wiggins columbia april 23, 2012 chris.wiggins@columbia.edu 4/23/12 Chris Wiggins • APAM: Department of Applied Physics and Applied Mathematics; • C2B2: Center for Computational Biology and Bioinformatics; • CISB: Columbia University Initiative in Systems Biology • ISDE: Institute for Data Sciences and Engineering Columbia University September 28, 2012
  • 2.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection thanks. . . - jake hofman (vbmod,vbfret) - jonathan bronson (vbfret) - jan-willem van de meent (hfret) - ruben gonzalez (vbfret, hfret) for more info: - vbfret.sourceforge.net - vbmod.sourceforge.net - hfret.sourceforge.net (soon) chris.wiggins@columbia.edu 4/23/12 BMC bioinformatics, 2010; PNAS 2009; Biophysical Journal 2009;
  • 3.
    bio+stats vbem/networks hierarchical 1biology and statistics genomics generative modeling 2 variational/biological networks variational Bayesian expectation maximization inference model selection 3 hierarchical/time series biological challenges inference model selection chris.wiggins@columbia.edu 4/23/12
  • 4.
    bio+stats vbem/networks hierarchicalgenomics generative modeling biology and statistics: chris.wiggins@columbia.edu 4/23/12
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    bio+stats vbem/networks hierarchicalgenomics generative modeling genomics: chris.wiggins@columbia.edu 4/23/12 plos comp bio `10, nyas `07, bmc bioinfo `06a, bmc bioinfo `06b, bioinfo `04, regulatory genomics 04, IEEE `05; NIPS (MLCB`03, `06 )
  • 6.
    bio+stats vbem/networks hierarchicalgenomics generative modeling generative modeling: chris.wiggins@columbia.edu 4/23/12 bmc bioinfo '10, PNAS`09, biophys j`09, PNAS`07, PNAS`06 NIPS (MLCB '09, '10, '11) IEEE sig. proc. `12; plos one `08,cell `07,prl `06, jcs `06, biophys j `06 pami `08, prl `08, NIPS (MLCB08), PNAS `05, bmc bioinfo `04
  • 7.
    bio+stats vbem/networks hierarchicalvbem inference model selection variational/biological networks: - variational bayesian expectation maximization - inference - model selection chris.wiggins@columbia.edu 4/23/12
  • 8.
    introduction formulation resultsextensions motivation history introduction: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Hartwell, Hopfield, Leibler and Murray NATURE|VOL 402 | SUPP | 2 DECEMBER 1999 | www.nature.com
  • 9.
    introduction formulation resultsextensions motivation history motivation: community detection in networks social networks biological networks problem: over-fitting/resolution limit chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 10.
    introduction formulation resultsextensions motivation history history: by community math/cs: spectral methods (Fiedler ’74, Shi + Malik ’00) math/cs: clustering generally (Taskar, Koller, Getoor) physics: modularity common thread: test w/ stochastic block model (’76, ’83) ergo: use as inference tool (Hastings 0604429, Newman+Liecht 061148) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 11.
    introduction formulation resultsextensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 12.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ
  • 13.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
  • 14.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max likelihood: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model:
  • 15.
    introduction formulation resultsextensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 16.
    introduction formulation resultsextensions generative model max likelihood max evidence algo formulation: generative model maximum likelihood maximum evidence complexity control. . . variational/mft. . . algorithm in physics: “test hamiltonian” in ML “variational bayesian methods” (Jordan, Mackay) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 17.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Increasing complexity
  • 18.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net http://research.microsoft.com/~minka/statlearn/demo/
  • 19.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 20.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net cf. “BIC” Schwartz, 1978
  • 21.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ
  • 22.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model: foreach node roll K-sided die with bias π to choose zi {1, . . . , K} foreach edge flip coin with bias ϑ+ if zi = zj , else ϑ− draw edge if coin lands heads up chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n
  • 23.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
  • 24.
    introduction formulation resultsextensions generative model max likelihood max evidence algo generative model. . . (bis) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Die rolling, coin flipping, and priors: where counts are: non-edges within modules edges within modules edges between modules non-edges between modules nodes in each module
  • 25.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max likelihood: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ
  • 26.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥)
  • 27.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Extends Newman (2004, 2006), Hastings (2006), Bornholdt & Reichardt (2004 & 2006) •Die rolling, coin flipping <-> infinite-range spin-glass Potts model: •Infer distributions over spin assignments, coupling constants, and chemical potentials and find number of occupied spin states H ⇥ ln p(A, ⌦z|⌦⇤, ⌦⇥) = i,j (JLAij JG) zi,zj + K µ=1 hµ N i=1 zi,µ JG ⇥ ln ⇥c/⇥d JL ⇥ ln(1 ⇥d)/(1 ⇥c) + JG hµ ⇥ ln µ p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥) Can do integrals, but sum is intractable, O(KN); use mean-field
  • 28.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Gibbs’/Jensen’s inequality (log of expected value bounds expected value of log) for any distribution q p(A|K) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ p(A,⌦z,⌦⇥, ⌦) = ⇥z ⇥ d⌦ ⇥ d⌦⇥ e H p(⌦)p(⌦⇥) Variational Bayes (MacKay, Jordan, Ghahramani, Jaakola, Saul 1999; cf. Feynman 1972)
  • 29.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A1): Beal, 2003
  • 30.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A2): Beal, 2003
  • 31.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net why would you do this? (A3): Beal, 2003
  • 32.
    introduction formulation resultsextensions generative model max likelihood max evidence algo max evidence: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Gibbs’/Jensen’s inequality (log of expected value bounds expected value of log) for any distribution q Variational Bayes (MacKay, Jordan, Ghahramani, Jaakola, Saul 1999; cf. Feynman 1972) • F is a functional of q; find approximation to posterior by optimizing approximation to evidence • Take q(z, π, θ)=q(z)q(π)q(θ); Qiμ is probability node i in module μ where expected counts are:
  • 33.
    introduction formulation resultsextensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net where expected counts are:
  • 34.
    introduction formulation resultsextensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net where expected counts are:
  • 35.
    introduction formulation resultsextensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 36.
    introduction formulation resultsextensions generative model max likelihood max evidence algo algo: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net suggests hard limit in step 3; sparse in step 1
  • 37.
    introduction formulation resultsextensions run time consistency good vs easy real data results: run time consistency required plot: good vs. easy real data karate biology american football chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 38.
    introduction formulation resultsextensions run time consistency good vs easy real data run time: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Main loop runtime for 104 nodes in MATLAB ~30 seconds
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    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 40.
    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 θ N=8, K=2, distribution after 2 iterations p(θ+ ) p(θ − )
  • 41.
    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • K=4? • Automatic complexity control: probability of occupation for extraneous modules goes to zero
  • 42.
    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • K=4? • Automatic complexity control: probability of occupation for extraneous modules goes to zero
  • 43.
    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net The “resolution limit” problem 1 2 3 4 5 57 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 58 59 60 1 2 3 4 5 57 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 58 59 60 Variational Bayes Girvan-Newman modularity
  • 44.
    introduction formulation resultsextensions run time consistency good vs easy real data consistency: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net The “resolution limit” problem 10 12 14 16 18 20 8 10 12 14 16 18 20 Ktrue K* K * =Ktrue Variational Bayes Modularity optimization 10 12 14 16 18 20 0.72 0.74 0.76 0.78 0.8 0.82 0.84 Ktrue GNmodularity Resolution limit problem on ring of 4−node cliques Single−clique communities (correct) Double−clique communities (incorrect) GN modularity (Clauset’s algorithm) Girvan-Newman modularity or Potts model w/ fixed parameters suffers from a resolution limit, where size of detected modules depends on network size Fortunato et. al. (2007), Kumpula et. al. (2007),
  • 45.
    introduction formulation resultsextensions run time consistency good vs easy real data good vs easy: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 46.
    introduction formulation resultsextensions run time consistency good vs easy real data real data: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Correctly infer K=12 conferences Validation: NCAA football schedule 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 nodes: teams edges: games shape: conference color: inferred module
  • 47.
    introduction formulation resultsextensions run time consistency good vs easy real data real data: chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net APS march meeting 2008 superconductivity (experimentalists) Nanotubes, Graphene superconductivity (theorists)
  • 48.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 49.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 50.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Nodes belong to “blocks” of varying size • Roll die for assignment of nodes to blocks • Probability of edge between two nodes depends only on block membership • Flip (one of K2) coins for edges • Result: mixture of Erdos-Renyi graphs 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2275 adjacency matrix
  • 51.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net vs
  • 52.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2803 adjacency matrix
  • 53.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net vs
  • 54.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0) running vbmod ... Elapsed time is 1.136925 seconds. running vbsbm ... Elapsed time is 1.398904 seconds. Fmod=13089.158019 Fsbm=13144.445782 vbmod wins
  • 55.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0.25) running vbmod ... Elapsed time is 1.557298 seconds. running vbsbm ... Elapsed time is 1.759527 seconds. Fmod=20457.142416 Fsbm=19457.306022 vbsbm wins
  • 56.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net >> vbsbm_vs_vbmod(0.5) running vbmod ... Elapsed time is 2.624886 seconds. running vbsbm ... Elapsed time is 1.440242 seconds. Fmod=26133.351210 Fsbm=23921.797625 vbsbm wins
  • 57.
    introduction formulation resultsextensions full SBM. . . probability of edge depends only on block membership: p(Aij |zi = µ, zj = ν) = ϑµν chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net • Using same framework we can compare the unconstrained and full stochastic block models via p(D|M,K*) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 perturbation to constrained model winpercentageforunconstrainedmodel 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 perturbation to constrained model winpercentageforunconstrainedmodel 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2100 adjacency matrix 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2048 adjacency matrix 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 2108 adjacency matrix
  • 58.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 59.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 60.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 61.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n
  • 62.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net Stochastic block models (Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987) i≠j zi zj Aij π θ c n L
  • 63.
    introduction formulation resultsextensions extensions: model extensions full SBM (done) hierarchical model p(Aij = 1|zi = zi ± 1) hierarchical modeling (ensemble of graphs) p(D|u, K) = ΠL i dϑi p(D|ϑi )p(ϑi |u, K) Rd embedding (latent are real) more ‘rigorous’ SOM? ‘correct’ for degree (allow variable affinity) (cf. Bader, Karrer) algorithm extensions BP (see earlier talks) map-reduce cvmod (model selection via cross validation) chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 64.
    introduction formulation resultsextensions for more info. . . code: MATLAB & python (inc. “full” SBM) (vbmod.sf.net) paper: arxiv 08 / prl 08 Hofman soon to come (not by me) code in C++, inc. full ‘vblabel propagation’ algo twitter-scale analysis chris.wiggins@columbia.edu 22.2.12 vbmod.sourceforge.net
  • 65.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hierarchical/time series: - biological challenges - inference - model selection chris.wiggins@columbia.edu 4/23/12
  • 66.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Jan-Willem van de Meent, Ruben Gonzalez, Chris Wiggins Columbia University
  • 67.
    Ramakrishnan et al– http://www.mrc-lmb.cam.ac.uk/ribo/
  • 68.
    Ramakrishnan et al– http://www.mrc-lmb.cam.ac.uk/ribo/
  • 69.
    Ramakrishnan et al– http://www.mrc-lmb.cam.ac.uk/ribo/
  • 70.
    Ramakrishnan et al– http://www.mrc-lmb.cam.ac.uk/ribo/
  • 71.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET = cy5 / (cy3 + cy5) Tinoco and Gonzalez, Genes Dev, 2011
  • 72.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Tinoco and Gonzalez, Genes Dev, 2011
  • 73.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009 (short-lived GS1 states correspond to an EF-G + GDPNP binding event)
  • 74.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009
  • 75.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 1. Identify states 2. Estimate Kinetic Rates 3. Average over many time series 4. Detect subpopulations
  • 76.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET Signal
  • 77.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
  • 78.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
  • 79.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram
  • 80.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 FRET SignalHistogram Idea: Find probability of belonging to each state
  • 81.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Expectation Maximization 1. calculate p(z | x, θi) 2. calculate θi+1 from p(z | x, θi)
  • 82.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Expectation Maximization 1. calculate p(z | x, θi) 2. calculate θi+1 from p(z | x, θi)
  • 83.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Learned Truth Accurate for occupancy of states, not so good for rate estimates
  • 84.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(x, z µ, σ, π) = p(x z, µ, σ)p(z π)
  • 85.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 probability of state depends on previous state p(zt+ =l zt =k) = Akl
  • 86.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 87.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 88.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 89.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(z =k) = πk
  • 90.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(zt+ =l zt =k) = Akl
  • 91.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 p(xt zt = k) = N(xt µk , σk)
  • 92.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Learned Real We’ve learned: parameters: θ = {µ, σ, π, A} states: p(z | x, θ)
  • 93.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2 States 3 States
  • 94.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Log-Evidence
  • 95.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Prior
  • 96.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Ensemble
  • 97.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Likelihood L = log p(x θ) = log z p(x, z θ) Log-Evidence best model has highest average likelihood L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Log-Evidence
  • 98.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Log-Evidence 31 L = log p(x u) = log z ∫ dθ p(x, z θ)p(θu) Lower Bound L = z ∫ dθ q(z)q(θ w)log p(x, z, θ u) q(z)q(θ w) ≥ log p(x u) q(z)q(θ w) p(z, θ x)
  • 99.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Lower bound tight for true posterior L = z ∫ dθ p(z, θ x)log p(x, z, θ u) p(z, θ x) = z ∫ dθ p(z, θ x)log[p(x u)] = log p(x u) L = log p(x u) − Dkl [q(z)q(θ w) p(z, θ x)]
  • 100.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 We’ve learned: parameters: q(θ | w) states: p(z | x, θ) δLn δq(zn) = VBEM Updates δLn δq(θn) =
  • 101.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 variability: photophysical/experimental
  • 102.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
  • 103.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
  • 104.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
  • 105.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31
  • 106.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Hierarchical Updates ∂ ∂u n Ln =
  • 107.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 31 Hierarchical Updates ∂ ∂u n Ln = “two-stage PEB model/CIHM”-Kass Steffey JASA 1989
  • 108.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2. Update p(θ | u) Until Σ Ln converges Until Ln converges • Update q(zn) • Update q(θn | wn) 1. Run VBEM on each trace δLn δq(zn) = Hierarchical UpdatesVBEM Updates δLn δq(θn) = ∂ ∂u n Ln =
  • 109.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 2. Update p(θ | u) Until Σ Ln converges Until Ln converges • Update q(zn) • Update q(θn | wn) 1. Run VBEM on each trace We’ve learned: p(θn, zn | xn) ≃ q(θn) q(zn) (for each trace) p(θ | u) (for ensemble)
  • 110.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 111.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 112.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 113.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 114.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 115.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 116.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12
  • 117.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Unbound EF-G bound Tinoco and Gonzalez, Genes Dev, 2011 Fei et al, PNAS, 2009
  • 118.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 ξntkl = p(znt = k, znt+ = l xn) 1. Run mixture model on posterior counts p(ξnA) = tkl Aξntkl kl p(ξn um) = ∫ dA p(Aum)p(ξn A) 2. Rerun with M x K block-diagonal form uA = uA uA uA M
  • 119.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 τfast τslow 2 4 8 16 32 64
  • 120.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 τfast τslow 2 4 8 16 32 64 reality
  • 121.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G Fei, Bronson, Hofman, Srinivas, Wiggins, Gonzalez, PNAS, 2009
  • 122.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G p(zk) ∼ e−Gk kB T log p(zk) − log p(zl ) = −(Gk − Gl )kBT + cst.
  • 123.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-G 50 nM EF-G 500 nM EF-G p(zk) ∼ e−Gk kB T Δ∆G=logit(p)
  • 124.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 no EF-Gbound fraction and life-times
  • 125.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 5 nM EF-Gbound fraction and life-times
  • 126.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 50 nM EF-Gbound fraction and life-times
  • 127.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 250 nM EF-Gbound fraction and life-times
  • 128.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 500 nM EF-Gbound fraction and life-times
  • 129.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection model selection: chris.wiggins@columbia.edu 4/23/12
  • 130.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, UnderfittedInf Out - Inf In
  • 131.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, CorrectOut vs In
  • 132.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 Low Noise, OverfittedInf Out - Inf In
  • 133.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, UnderfittedInf Out - Inf In
  • 134.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, CorrectInf Out - Inf In
  • 135.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, OverfittedInf Out - Inf In
  • 136.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 High Noise, OverfittedInf Out - Inf In
  • 137.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection hfret. . . chris.wiggins@columbia.edu 4/23/12 the future, in progress: X
  • 138.
    bio+stats vbem/networks hierarchicalbiological challenges inference model selection thanks. . . - jake hofman (vbmod,vbfret) - jonathan bronson (vbfret) - jan-willem van de meent (hfret) - ruben gonzalez (vbfret, hfret) for more info: - vbfret.sourceforge.net - vbmod.sourceforge.net - hfret.sourceforge.net (soon) chris.wiggins@columbia.edu 4/23/12 BMC bioinformatics, 2010; PNAS 2009; Biophysical Journal 2009;
  • 139.
    traditional role ofstatistics in biophysics “if your experiment needs statistics, you ought to have done a better experiment” -lord rutherford