On a Class of Genetic Genealogical Tree Models P. DEL MORAL Lab. - PowerPoint PPT Presentation

Numerical and Stochastic Models, Paris Oct. 2006 On a Class of Genetic Genealogical Tree Models P. DEL MORAL Lab. J.A. Dieudonn´ e, Univ. Nice Sophia Antipolis, France ֒ → F-K Formulae, Genealogical and IPS , Springer (2004) + References therein → (delmoral@unice.fr) ֒ → [preprints+info.] http://math1.unice.fr/ delmoral/ ֒ 1

Introduction • Evolutionary models and Feynman-Kac formulae • Genetic genealogical models and Feynman-Kac limiting measures • Application model areas : particle physics (absorbing medium, ground states), biology (polymers, macromolecules), statistics (particle simulation, restricted Markov, target distributions), sare event analysis (importance sampling, multilevel branching), signal processing, filtering. • Asymptotic analysis : ֒ → Functional representations ≃ precise propagations of chaos expansions . (joint work F. Patras, S. Rubenthaler) – Combinatorial differential calculus – Permutation group analysis of (colored) forests (wreath product of permutation groups, Hilbert series techniques,. . . ) Discrete time models � Continuous time version = Moran type genetic models ( ∼ joint works with L. Miclo, see also [PhD ⊕ articles] M. Rousset) 2

Evolutionary type models Simple Genetic Branching Algo. Mutation Selection/Branching Metropolis-Hastings Algo. Proposal Acceptance/Rejection Sequential Monte Carlo methods Sampling Resampling (SIR) Filtering/Smoothing Prediction Updating/Correction Particle ∈ Absorbing Medium Evolution Killing/Creation/Anhiling Other Botanical Names: multi-level splitting (Khan-Harris 51), prune enrichment (Rosenbluth 1955), switching algo. (Magill 65), matrix reconfiguration (Hetherington 84), restart (Villen-Altamirano 91), particle filters (Rigal-Salut-DM 92), SIR filters (Gordon-Salmon-Smith 93, Kitagawa 96), go- with-the-winner (Vazirani-Aldous 94), ensemble Kalman-filters (Evensen 1994), quantum Monte Carlo methods (Melik-Nightingale 1999), sequential Monte Carlo Methods (Arnaud Doucet 2001), spawning filters (Fisher-Maybeck 2002), SIR Pilot Exploration Resampling (Liu-Zhang 2002),... 3

⇐ ⇒ Particle Interpretations of Feynman-Kac models Since R. Feynman’s phD. on path integrals 1942 Physics ← → Biology ← → Engineering Sciences ← → Probability/Statistics • Physics : – FKS ∈ nonlinear integro-diff. ´ eq. ( ∼ generalized Boltzmann models). – Spectral analysis of Schr¨ odinger operators and large matrices with nonnegative entries. (particle evolutions in disordered/absorbing media) – Multiplicative Dirichlet problems with boundary conditions. – Microscopic and macroscopic interacting particle interpretations. • Biology : – Self-avoiding walks, macromolecular polymerizations. – Branching and genetic population models. – Coalescent and Genealogical evolutions. 4

• Rare events analysis : – Multisplitting and branching particle models (Restart). – Importance sampling and twisted probability measures. – Genealogical tree based simulation methods. • Advanced Signal processing : – Optimal filtering/smoothing/regulation, open loop optimal control. – Interacting Kalman-Bucy filters. – Stochastic and adaptative grid approximation-models • Statistics/Probability : – Restricted Markov chains (w.r.t terminal values, visiting regions,...) – Analysis of Boltzmann-Gibbs type distributions (simulation, partition functions,...). – Random search evolutionary algorithms, interacting Metropolis/simulated annealing algo.

Simple Genetic evolution/simulation models − → only 2 ingredients!! (Discrete time parameter n ∈ N = { 0 , 1 , 2 , ... } , state spaces E n ( ∈ { Z d , R d , R d × . . . × R d , ... } ) � �� � ( n +1) − times • Mutation/exploration/prediction/proposal : → Markov transitions M n ( x n − 1 , dx n ) from E n − 1 into E n . • Selection/absorption/updating/acceptance : → Potential functions G n from E n into [0 , 1]. 5

A Genetic Evolution Model ⇒ Markov chain ξ n = ( ξ 1 n , . . . , ξ N n ) ∈ E N n = E n × . . . × E n � �� � N − times selection mutation → � ξ n ∈ E N ξ n ∈ E N → ξ n +1 ∈ E N − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − n n n +1 • Selection transition ( ∃ � = types → Ex.: accept/reject) n � � ξ i ξ i n = ξ i G n ( ξ i with proba. n ) [Acceptance] n Otherwise we select a better fitted individual in the current configuration N � � ξ i n = ξ j G n ( ξ j G n ( ξ k with proba. n ) / n ) [Rejection + Selection] n k =1 • Mutation transition n , . ) � n +1 ∼ M n +1 ( � ξ i n � ξ i ξ i Continuous time models : ( M, G ) = ( Id + ∆ L, e − V ∆ ) � L -motions ⊕ expo. random clocks rate V +Uniform selection 6

A Genealogical tree model Important observation [Historical process] X ′ n ∈ E ′ Markov chain n ⇓ X n = ( X ′ 0 , . . . , X ′ n ) ∈ E n = ( E ′ 0 × . . . × E ′ n ) Markov chain ∈ path spaces → Markov transitions M n ( x n − 1 , dx n ) [elementary extensions] X n +1 = (( X ′ 0 , . . . , X ′ n ) , X ′ n +1 ) = ( X n , X ′ n +1 ) 7

Genetic Evolution Model on Path Spaces=Genealogical tree model X n = ( X ′ 0 , . . . , X ′ G n ( X n ) = G ′ n ( X ′ n ) M n n ) Markov transitions and ↓ Genetic path-valued particle Model � ξ i ( ξ i 0 ,n , ξ i 1 ,n , . . . , ξ i = n,n ) n � ( � 0 ,n , � 1 ,n , . . . , � ξ i ξ i ξ i ξ i n,n ) ∈ E n = ( E ′ 0 × . . . × E ′ = n ) n • Path acceptance/(rejection+selection). • Path mutation = path elementary extensions. 8

Occupation/Empirical measures ( ∀ f n test function on E n ) N N � � n ( f n ) = 1 n ) = 1 η N f n ( ξ i f n ( ξ i 0 ,n , ξ i 1 ,n , . . . , ξ i n,n ) � �� � N N i =1 i =1 i -th ancestral lines ↓ Unbias-particle measures & Unnormalized Feynman-Kac measures : � � γ N η N η N n ( f n ) = n ( f n ) × p ( G p ) − → N →∞ γ n ( f n ) = E ( f n ( X n ) G p ( X p )) 0 ≤ p<n 0 ≤ p<n Notes: n (1) = � → N →∞ γ n (1) = E ( � • f n = 1 ⇒ γ N 0 ≤ p<n η N p ( G p ) − 0 ≤ p<n G p ( X p )) • Path-space models � [ X n = ( X ′ 0 , . . . , X ′ n ) and G n ( X n ) = G ′ n ( X ′ n ) ] ⇒ γ n ( f n ) = E ( f n ( X ′ 0 , . . . , X ′ G ′ p ( X ′ n ) p )) 0 ≤ p<n 9

= ⇒ Occupation measure & Normalized Feynman-Kac measures: N � 1 η N f n ( ξ i n ) = γ N n ( f n ) /γ N n ( f n ) = n (1) − → N →∞ η n ( f n ) = γ n ( f n ) /γ n (1) N i =1 Path-space models [ X n = ( X ′ 0 , . . . , X ′ n ) and G n ( X n ) = G ′ n ( X ′ n ) ] ⇓ n ) � E ( f n ( X ′ 0 , . . . , X ′ 0 ≤ p<n G ′ p ( X ′ p )) E ( � η n ( f n ) = 0 ≤ p<n G ′ p ( X ′ p )) Note: � � − γ N n ( f n ) = η N η N γ n ( f n ) = η n ( f n ) × η p ( G p ) ( ← n ( f n ) × p ( G p )) 0 ≤ p<n 0 ≤ p<n

Applications : • Particle physics (absorbing medium, ground states) • Biology (polymers, macromolecules) • Statistics (particle simulation, restricted Markov, target distributions) • Rare event analysis (importance sampling, multilevel branching) • Signal processing, filtering → [Stochastic Engineering + Scilab Prog. (Master 2) ( ⇒ in french)] ֒ http://math1.unice.fr/ delmoral/eng.html ֒ → [IRISA ASPI A ppl. S tat. Syst. de P articules en I nteraction] http://www.irisa.fr/activites/equipes/aspi 10

Particle physics: Markov X n ∈ Absorbing medium G ( x ) = e − V ( x ) ∈ [0 , 1] absorption exploration n ∈ E c = E ∪ { c } → � X c X c → X c − − − − − − − − − − − − − − − − − − − − − − − − n n +1 → � n ); otherwise the particle is killed and � X c n = X c n , with proba G ( X c X c Absorption/killing: − n = c . ⇓ { x : G ( x ) = 0 } − → Hard obstacles A = inf { n ≥ 0 ; � T + n = � X c X c X c = n = c } − → Absorption time T + n = c T γ n = Law( X c = ⇒ Feynman-Kac models ( G, X n ) : n ; T ≥ n ) and γ n (1) = Proba( T ≥ n ) ⇓ η n = Law( X c n | T ≥ n ) = Law(( X ′ c 0 , . . . , X ′ c n ) | T ≥ n ) 11

Lyapunov exponents and grounds states ( X n ∼ M , G ∈ [0 , 1], T abs. time) ( ⊕ Schrodinger op. L+V � joint work with L. Miclo ESAIM 2003, see also M. Rousset PhD.+articles ) � � η p ( G ) ≃ e − λn Proba( T ≥ n ) = E ( λ = − log η ∞ ( G ) G ( X p )) = with 0 ≤ p<n 0 ≤ p<n µ − reversible ⇒ η ∞ ( f ) = µ ( H M ( f )) M with GM ( H ) = λH µ ( H ) and Ψ( η ∞ )( f ) := η ∞ ( Gf ) η ∞ ( G ) = µ ( Hf ) µ ( H ) N-particle approx. : � � η p ( G ) = γ n (1) ≃ n e − λn γ N η N n (1) = p ( G ) ≃ N 0 ≤ p<n 0 ≤ p<n Ψ( η n ) ≃ n Ψ( η ∞ ) = µ ( Hf ) Ψ( η N n ) ≃ N µ ( H ) 12

Biology: Macromolecules and Directed Polymers • Self avoiding walks X ′ n ∈ Z d X n = ( X ′ 0 , . . . , X ′ n − 1 } ( X ′ n ) and G n ( X n ) = 1 �∈{ X ′ n ) 0 ,...,X ′ γ n (1) = Proba( ∀ 0 ≤ p � = q ≤ n, X ′ p � = X ′ η n = Law( X ′ 0 , . . . , X ′ n | ∀ 0 ≤ p � = q ≤ n, X ′ p � = X ′ q ) and q ) Ex. : ( Connectivity constants ) (2 d ) n × # { non intersecting walks with length n } ≃ exp ( c n ) 1 γ n (1) = (2 d ) n • Edwards’ model � X n = ( X ′ 0 , . . . , X ′ p ( X ′ n ) and G n ( X n ) = exp {− β 1 X ′ n ) } 0 ≤ p<n 13