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Introduction Gibbs sampling algorithm Case studies Gibbs sampling for parsimonious Markov models with latent variables Ralf Eggeling 1 , Pierre-Yves Bourguignon 2 , Andr´ e Gohr 1 , Ivo Grosse 1 1 Martin Luther University Halle-Wittenberg 2 Max Planck Institute for Mathematics in the Sciences Max Planck Institute for Mathematics in the Sciences Eggeling et al. Gibbs sampling for parsMMs with latent variables 1

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA CA GA Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA CA TA GA Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA CA TA AC GA AG AT GC GG GT Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA CA TA AC CC GA AG CG AT CT GC TC GG TG GT TT Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 1: Parsimonious Markov models proposed by Bourguignon (2008) generalize variable order Markov models use parsimonious context trees (PCTs) here: inhomogeneous models → seperate PCTs for each random variable A A A A A A C,G,T C,G,T C,G,T C,G,T C,G,T C,G,T A A A A A A C,G C,G C,G C,G C,G C,G T T T T T T A,G A,G A,G A,G A,G A,G C,T C,T C,T C,T C,T C,T AA CA TA AC CC GA AG CG AT CT GC TC GG TG GT TT Eggeling et al. Gibbs sampling for parsMMs with latent variables 2

Introduction Gibbs sampling algorithm Case studies Premise 2: Latent variable models many practical applications: latent variables, unobserved/missing data Eggeling et al. Gibbs sampling for parsMMs with latent variables 3

Introduction Gibbs sampling algorithm Case studies Premise 2: Latent variable models many practical applications: latent variables, unobserved/missing data examples: Naive Bayes Hidden Markov models Mixture models Eggeling et al. Gibbs sampling for parsMMs with latent variables 3

Introduction Gibbs sampling algorithm Case studies Premise 2: Latent variable models many practical applications: latent variables, unobserved/missing data examples: Naive Bayes Hidden Markov models Mixture models Mixture models model assumption: data point i generated from one out of C component models → latent variable u i ∈ { 1 , . . . , C } analytical learning infeasible approximative algorithms: EM algorithm Gibbs sampling Eggeling et al. Gibbs sampling for parsMMs with latent variables 3

Introduction Gibbs sampling algorithm Case studies Premise 3: Bayesian prediction Classical prediction estimate optimal parameters ˆ Θ( X ) from training data X P classic ( Y | X ) = P ( Y | ˆ Θ( X )) Eggeling et al. Gibbs sampling for parsMMs with latent variables 4

Introduction Gibbs sampling algorithm Case studies Premise 3: Bayesian prediction Classical prediction estimate optimal parameters ˆ Θ( X ) from training data X P classic ( Y | X ) = P ( Y | ˆ Θ( X )) Bayesian prediction do not estimate optimal parameters � P Bayes ( Y | X ) = P ( Y | Θ) P (Θ | X ) d Θ Eggeling et al. Gibbs sampling for parsMMs with latent variables 4

Introduction Gibbs sampling algorithm Case studies Premise 3: Bayesian prediction Classical prediction estimate optimal parameters ˆ Θ( X ) from training data X P classic ( Y | X ) = P ( Y | ˆ Θ( X )) Bayesian prediction do not estimate optimal parameters � P Bayes ( Y | X ) = P ( Y | Θ) P (Θ | X ) d Θ classical prediction approximates Bayesian prediction posterior concentrated around ˆ Θ → good approximation posterior diverse → bad approximation Eggeling et al. Gibbs sampling for parsMMs with latent variables 4

Introduction Gibbs sampling algorithm Case studies Putting premises together Parsimonious Markov models Gibbs Latent variable Sampling models Bayesian prediction Eggeling et al. Gibbs sampling for parsMMs with latent variables 5

Introduction Gibbs sampling algorithm Case studies Gibbs sampling algorithm goal: sample from posterior distribution Gibbs sampling: sample iteratively from conditional probability distributions of each variable/parameter probability parameters (2) tree structures (1) latent variables (3) probability parameters → simple latent variables → simple structure → difficult Eggeling et al. Gibbs sampling for parsMMs with latent variables 6

Introduction Gibbs sampling algorithm Case studies Structure sampling probability of a PCT structure: κ B ( � N w + � α w ) � P ( τ | X ) ∝ B ( � α w ) w ∈C τ product of leaf scores observation: subtree (red) probability independent of sibling subtree(s) given subtree root (green) A C,G,T A C,G T A,G C,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 7

Introduction Gibbs sampling algorithm Case studies Structure sampling dynamic programming on extended PCT → sibling nodes form P ( A ) \ { ∅ } X A C G T A,C A,G A,T C,G C,T G,T A,C,G A,C,T A,G,T C,G,T A,C,G,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 8

Introduction Gibbs sampling algorithm Case studies Structure sampling dynamic programming on extended PCT → sibling nodes form P ( A ) \ { ∅ } depth identical to that of PCT X A C G T A,C A,G A,T C,G C,T G,T A,C,G A,C,T A,G,T C,G,T A,C,G,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 8

Introduction Gibbs sampling algorithm Case studies Structure sampling dynamic programming on extended PCT → sibling nodes form P ( A ) \ { ∅ } depth identical to that of PCT traverse tree top-down X A C G T A,C A,G A,T C,G C,T G,T A,C,G A,C,T A,G,T C,G,T A,C,G,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 8

Introduction Gibbs sampling algorithm Case studies Structure sampling sample subtrees bottom-up child nodes are a) leaves b) roots of valid PCT subtrees X A A C G T A,C A,G A,T C,G C,T G,T A,C,G A,C,T A,G,T C,G,T A,C,G,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 9

Introduction Gibbs sampling algorithm Case studies Structure sampling sample subtrees bottom-up child nodes are a) leaves b) roots of valid PCT subtrees compute score of all valid child combinations (15) X A A C G T A,C A,G A,T C,G C,T G,T A,C,G A,C,T A,G,T C,G,T A,C,G,T Eggeling et al. Gibbs sampling for parsMMs with latent variables 9