An introduction to particle simulation of rare events P. Del Moral Centre INRIA de Bordeaux - Sud Ouest Post-graduate course on ”Advanced Optimisation Techniques”, CSC Doctorate School. Luxembourg University Lecture 3 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 1 / 36
Outline Introduction, motivations 1 Particle interpretations of Feynman-Kac models 2 Additive functionals 3 Normalizing constant estimation 4 Some references 5 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 2 / 36
Summary 1 Introduction, motivations Some rare event problems Stochastic models Importance sampling techniques The heuristic of particle methods 3 types of occupation measures Particle interpretations of Feynman-Kac models 2 Additive functionals 3 Normalizing constant estimation 4 Some references 5 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 3 / 36
Rare event analysis Stochastic process X ⊕ Rare event A : Proba ( X ∈ A ) & Law (( X 0 , . . . , X t ) | X ∈ A ) ⊃ engineering/physics/biology/economics/finance : Finance : ruin/default probabilities, financial crashes, eco. crisis,... engineering : networks overload, breakdowns, engines failures,... Physics : climate models, directed polymer conformations, particle in absorbing medium, ground states of Schroedinger models. Statistics : tail probabilities, extreme random values. Combinatorics : Complex enumeration problems. Process strategies ∈ Rare event ⇒ Control and prediction. X t = F t ( X t − 1 , W t ) → Law (( W 0 , . . . , W t ) | X ∈ A ) P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 4 / 36
Stochastic models Only 2 Ingredients 1 Physical/biological/financial process : queuing network, portfolio, volatility process, stock market evolutions, interacting/exchange economic models ... 1 Potential function (energy type, indicator, restrictions): critical level crossing, penalties functions, constraints subsets, performance levels, long range dependence... Objectives Estimation of the probability of the rare event. Computing the full distributions of the path of the process evolving in the critical regime � prediction ⊕ control. P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 5 / 36
Twisted Monte Carlo methods P ( X ∈ A ) = 10 − 10 � Find Q s.t. Q ( A ) ≃ 1 Elementary Monte Carlo estimate X i iid Q � � d P d Q ( x ) 1 A ( x ) Q ( dx ) ≃ P N ( A ) := 1 d P d Q ( X i ) 1 A ( X i ) P ( A ) := N 1 ≤ i ≤ N � d P Variance ≃ (ex. cross-entropy : Q ∈ { Q a , a ∈ A} ) d Q ( x ) 1 A ( x ) P ( dx ) Drawbacks Huge variance if Q badly chosen � optimal choice Q ( dx ) ∝ 1 A ( x ) P ( dx ). Need to twist the original reference process X . Stochastic evolution X = ( X 0 , . . . , X n ) n � d P n p k ( X k | X k − 1 ) ( X ) := degenerate product martingale w.r.t. n d Q n q k ( X k | X k − 1 ) k =0 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 6 / 36
The heuristic of particle methods Flow of measure with increasing sampling complexity Rare event = cascade/series of intermediate less-rare events ( ↑ energy levels, physical gateways, index level crossings). Conditional probability flow = flow of optimal twisted measures n → η n = Law(process | series of n intermediate events) Rare event probabilities= Normalizing constants. Particle methods (Sampling a genealogical type default tree model ⊕ % success or default) Explorations/Local search propositions of the solution space. Branching-Selection individuals ∈ ↑ critical regimes. P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 7 / 36
5 Examples of flow of target measures η n = Loi (( X 0 , . . . , X n ) | ∀ 0 ≤ p ≤ n X p ∈ A p ) 1 η n ( dx ) ∝ e − β n V ( x ) λ ( dx ) with β n ↑ 2 η n ( dx ) ∝ 1 A n ( x ) λ ( dx ) with A n ↓ 3 η n = Loi K π (( X 0 , . . . , X n ) | X n = x n ). 4 η n = Loi ( X hits B n | X hits B n before A ) 5 5 particle heuristics : M n -local moves ⊕ individual selections ∈ A n i.e. ∼ G n = 1 A n 1 MCMC local moves η n = η n M n ⊕ individual selections ∝ G n = e − ( β n +1 − β n ) V 2 MCMC local moves η n = η n M n ⊕ individual selections ∝ G n = 1 A n +1 3 M -local moves ⊕ Selection G ( x 1 , x 2 ) = π ( dx 2 ) K ( x 2 , dx 1 ) 4 π ( dx 1 ) M ( x 1 , dx 2 ) M n -local moves ⊕ Selection-branching on upper/lower levels B n . 5 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 8 / 36
� � � � � Interaction/branch. process ֒ → 3 types of occupation measures ( N = 3) � • • = • • • � � � � � � � � • � • • = • • � � � � � � � � � � � � � � � � • • = • • • N � → 1 Current population ֒ δ ξ i n ← i -th individual at time n N i =1 N � → 1 Genealogical tree model ֒ δ ( ξ i n , n ) ← i -th ancestral line 0 , n ,ξ i 1 , n ,...,ξ i N i =1 � N → 1 Complete genealogical tree model ֒ i =1 δ ( ξ i n ) 0 ,ξ i 1 ,...,ξ i N N � → 1 G n ( ξ i ⊕ Empirical mean potentials [success % ( G n = 1 A ) ] ֒ n ) N i =1 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 9 / 36
Equivalent Stochastic Algorithms : Genetic and evolutionary type algorithms. Spatial branching models. Sequential Monte Carlo methods. Population Monte Carlo models. Diffusion Monte Carlo (DMC), Quantum Monte Carlo (QMC), ... Some botanical names ∼ � = application domain areas : bootsrapping, selection, pruning-enrichment, reconfiguration, cloning, go with the winner, spawning, condensation, grouping, rejuvenations, harmony searches, biomimetics, splitting, ... � 1950 ≤ [(Meta)Heuristics] ≤ 1996 ≤ Feynman-Kac mean field particle model P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 10 / 36
Summary 1 Introduction, motivations Particle interpretations of Feynman-Kac models 2 Quelques notations Asymptotic Analysis Nonlinear Markov chains Mean field particle interpretations Some cv. results Additive functionals 3 Normalizing constant estimation 4 Some references 5 P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 11 / 36
Some notation E measurable state space, P ( E ) proba. on E , B ( E ) bounded meas. functions � ( µ, f ) ∈ P ( E ) × B ( E ) − → µ ( f ) = µ ( dx ) f ( x ) M ( x , dy ) integral operator over E � M ( f )( x ) = M ( x , dy ) f ( y ) � [ µ M ]( dy ) = µ ( dx ) M ( x , dy ) (= ⇒ [ µ M ]( f ) = µ [ M ( f )] ) Bayes-Boltzmann-Gibbs transformation : G : E → [0 , ∞ [ with µ ( G ) > 0 1 Ψ G ( µ )( dx ) = µ ( G ) G ( x ) µ ( dx ) ⇒ Matrix notations µ = [ µ (1) , . . . , µ ( d )] and f = [ f (1) , . . . , f ( d )] ′ E finite ⇐ P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 12 / 36
Infinite Population N ↑ ∞ ”=” Feynman-Kac measures ≃ ( G n , M n ) N � n ( f ) := 1 → N ↑∞ η n ( f ) := γ n ( f ) η N f ( ξ i n ) − N γ n (1) i =1 with the un-normalized measures : � f n ( X n ) γ n ( f ) := E G p ( X p ) 0 ≤ p < n [Potential functions G n ] & [ X n Markov chain ∼ transitions M n ] Feynman-Kac models ⊃ ALL of the above heuristics G n = 1 A n , e − ( β n − β n − 1 ) V , Metropolis ratio, level crossing detect.,... ⊕ (LDP)Twisted meas. : P ( V n ( X n ) ≥ a ) & Loi (( X 0 , ..., X n ) | V n ( X n ) ≥ a ) e λ V n ( X n ) P ( X n ∈ . ) � G n ( X n − 1 , X n ) = e λ [ V n ( X n ) − V n − 1 ( X n − 1 )] X n = ( X ′ 0 , . . . , X ′ U p ( X ′ n ) � V n ( X n ) = sup p ) 0 ≤ p ≤ n P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 13 / 36
A first ”detailed” example Boltzmann-Gibbs Measures : η n ( dx ) = 1 e − β n V ( x ) λ ( dx ) Z n Feynman-Kac representation : � η n ( f ) := γ n ( f ) f n ( X n ) e − ( β p +1 − β p ) V ( X p ) with γ n ( f ) := E γ n (1) 0 ≤ p < n and P ( X n ∈ dx n | X n − 1 = x n − 1 ) = M n ( x n − 1 , dx n ) with η n = η n M n Note : � e − β n V � Z n = λ � e − ( β n − β n − 1 ) V e − β n − 1 V � � e − ( β p +1 − β p ) V � � λ ( β 0 =0) × Z n − 1 = = η p λ ( e − β n − 1 V ) � �� � 0 ≤ p < n η n − 1 ( e − ( β n − β n − 1) V ) P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 14 / 36
A second ”detailed” example Restriction of measures : A n ↓ (Ex.: A n = [ a n , ∞ [ � tails probab.) η n ( dx ) = 1 1 A n ( x ) λ ( dx ) Z n Feynman-Kac representation : � η n ( f ) := γ n ( f ) f n ( X n ) with γ n ( f ) := E 1 A p +1 ( X p ) γ n (1) 0 ≤ p < n and P ( X n ∈ dx n | X n − 1 = x n − 1 ) = M n ( x n − 1 , dx n ) with η n = η n M n Note : Z n = λ ( A n ) � � � λ 1 A n 1 A n − 1 � � ( A 0 = E ) = � � × Z n − 1 = η p 1 A p +1 λ 1 A n − 1 0 ≤ p < n � �� � η n − 1 (1 An ) P. Del Moral (INRIA) INRIA Bordeaux-Sud Ouest 15 / 36
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