Symposium, Univ. Bonn MDV Sept. 2006 Stochastic Algorithms and Markov Processes Feynman-Kac particle models Coalescent tree based functional representations P. DEL MORAL, F. PATRAS, S. RUBENTHALER Lab. J.A. Dieudonn´ e, Univ. Nice Sophia Antipolis, France → Coalescent tree based functional representations for some Feynman-Kac ֒ particle models https://hal.ccsd.cnrs.fr/ccsd-00086532 ֒ → (delmoral@math.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 • Functional representations ≃ precise propagations of chaos expansions . – Combinatorial differential calculus – Permutation group analysis of (colored) forests (wreath product of permutation groups, Hilbert series techniques,. . . ) • (Applications). 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 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
Motivating example → filtering/hidden Markov chains/Bayesian Stat. Signal process X n = Markov chain ∈ E n Observation/Sensor eq. Y n = H n ( X n , V n ) ∈ F n with P ( H n ( x n , V n ) ∈ dy n ) = g n ( x n , y n ) λ n ( dy n ) Example: Y n = h n ( X n ) + V n ∈ F n = R , with Gaussian noise V n = N (0 , 1) ⇓ P ( h n ( x n ) + V n ∈ dy n ) = (2 π ) − 1 / 2 e − 1 2 ( y n − h n ( x n )) 2 dy n = exp [ h n ( x n ) y n − h 2 N (0 , 1)( dy n ) n ( x n ) / 2] � �� � � �� � λ n ( dy n ) g n ( x n ,y n ) Prediction/filtering/smoothing → Feynman-Kac representation G n ( x n ) = g n ( x n , y n ) η n = Law( X n | Y 0 = y 0 , . . . , Y n − 1 = y n − 1 ) = Law( X ′ 0 , . . . , X ′ n | Y 0 = y 0 , . . . , Y n − 1 = y n − 1 ) 10
Rather complete asymptotic theory ( n, N ) → ∞ (usual LLN, CLT, LDP,...) → F-K Formulae, Genealogical and IPS , Springer (2004) + References therein ֒ Some examples: • Weak convergence [ p ≥ 1 + F n not too large + regular mutations] (JTP 2000, joint work with M. Ledoux) √ n ( f n ) − η n ( f n ) | p ) 1 /p ≤ c ( p ) / | η N sup E ( sup N n ≥ 0 f n ∈F n √ n (1 ] −∞ ,x ] ) − η n (1 ] −∞ ,x ] ) | p ) 1 /p ≤ c ( p ) / | η N Ex : E n = R , F n = { 1 ] −∞ ,x ] ; x ∈ R } ⇒ sup E (sup N n ≥ 0 x ∈ R • Propagation-of-chaos estimates [ q ≤ N finite block size] (TVP+SIAM PTA 2006, joint work with A. Doucet) + 1 P N n,q := Law( ξ 1 n , . . . , ξ q n ) ≃ η ⊗ q N ∂ 1 P n,q ∂ 1 P n,q � ∂ 1 P n,q � tv ≤ c q 2 with signed meas. s.t. sup n n ≥ 0 11
Problem : Pb : Find a functional representation at any order? + 1 N ∂ 1 P n,q + . . . + 1 1 P N n,q ≃ η ⊗ q N k ∂ k P n,q + N k +1 ∂ k P N n n,q with a bounded remainder measure sup N ≥ 1 � ∂ k +1 P N n,q � tv < ∞ Consequences : • Sharp + strong propagations of chaos estimates at any order. • Wick product formulae on forests. • Sharp L p -mean error bounds. • Law of large numbers for U -statistics for interacting processes. • . . . 12
Tensor product measures � � n ) ⊗ q = 1 1 n ) ⊙ q = ( η N ( η N and δ ξ a δ ξ a � N q n ( N ) q n a ∈ [ N ] [ q ] a ∈� q,N � with ( ξ a (1) , . . . , ξ a ( q ) ξ a := ) n n n N q mappings [ q ] := { 1 , . . . , q } � [ N ] := { 1 , . . . , N } ; [ N ] [ q ] := � q, N � := ( N ) q := N ! / ( N − q )! one-to-one mappings 13
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