MCMC チュートリアル - ISM

1 mcmc Q A mcmc ) mcmc mcmc 1950s~ 1990s~ ( 1980s) 1990s~ Next ? QMCMC A mcmc ( mcmc Stan JAGS data {int T;int Y[T];int Z;}model {for(t in 3:T)beta[t] ~ normal(beta[t-1], s_mu);for(t in 1:T)Y[t]~poisson(Z* prob[t]);}transformed parameters {real prob[T];for(t in 1:T)prob[t,1]<-exp(beta[t]); }parameters {real beta[T];real<lower=0> s_mu.}

2 }STAN R,Python Q A @ simulated annealing)@ @ mcmc ( burn in ) (detailed balance) mcmc GibbsSampler MH ) Stan )

3 1step Hamilton emcee mcmc rnd rnd W (6 6 1500 CT mcmc ! data {int T;int Y[T];int Z;}model {for(t in 3:T)beta[t] ~ normal(beta[t-1], s_mu);for(t in 1:T)Y[t]~poisson(Z* prob[t]);}transformed parameters {real prob[T];for(t in 1:T)prob[t,1]<-exp(beta[t]); }parameters {real beta[T];real<lower=0> s_mu.)}

4 }STAN R,Python Example 1 Paralell Tempering (Replica Exchange mcmc )Kitajima & Kikuchi (2015) 10 2056 mcmc x U(x)=0 rsum_i = i csum_j = j dsum_k = ( ) jik=2k=1 U=0 Complete Magic SquareU(x) jik=2k=1 x U(x)=0 mcmc U U U=0 U=0 Entropic Sampling Histograma constant, [ ] 12347586(1) Request of a broad flat region (2) Insufficent Sampling in each iteration instabilityAn Example of Bad ConvergenceWang-Landau Weights : Set.

5 Update with the the current state isand discount: increment: becomes sufficiently flat and Goto 2 else Goto become sufficiently small then 2 Chaotic Dynamical SystemsSearch for rare initial conditions that gives rare , Coupled Standard MapProbability of regular trajectory (fragments) embedded in chaotic seaOur method Initial Condition Sampling (deterministic systems) Muticanonical (or Parallel Tempering) efficient sampling with correct Mixing Time Calculate the distributioninitial conditionMulticanonical SamplingSimulation of the dynamical systemMetropolis-HastingsiterationsOptim ization of Weightby Wang-Landau (or entropic sampling) iterationsResults(Figures)

6 See of Rare Trajectories in Chaotic Dynamical SystemsAkimasaKitajima, Yukito IbaComputer Physics CommunicationsVolume 182, Issue 1, Pages 1-280 (January 2011)