Hamiltonian Monte Carloによるベイズ推定ソフトウェアStanの紹介

Hamiltonian Monte Carlo Stan 2014-07-28 4

Hamiltonian Monte Carlo (HMC)! Stan

Hamiltonian Monte Carlo MCMC (Markov chain Monte Carlo) Hybrid Monte Carlo MCMC + deterministic simulation method!

MCMC Metropolis-Hastings 0 +/

HMC 0 +/

http://mc-stan.org/

Stan Sampling through adaptive neighborhoods (Gelman et al. 2013)

Stanisaw Ulam (19091984) http://en.wikipedia.org/wiki/Stanislaw_Ulam#mediaviewer/File:STAN_ULAM_HOLDING_THE_FERMIAC.jpg

Stan 20128 1.0 201212 1.1 20133 1.2 201310 2.0 201312 2.1 20142 2.2 20146 2.3 20147 2.4

Stan 3 RStan! R! PyStan! Python! CmdStan!

Stan Hamiltonian Monte Carlo, No-U-Turn Sampler C++

BUGS HMC

int real vector row_vector matrix

*1010.3 BUGS model! {! for (i in 1:N) {! Y[i] ~ dbin(q[i], 8)! logit(q[i]) # read data! > d ! 1010.3

> model

> data ! > fit print(fit, digits = 2)! Inference for Stan model: model.! 3 chains, each with iter=10100; warmup=100; thin=10; ! post-warmup draws per chain=1000, total post-warmup draws=3000.! ! mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat! beta 0.04 0.01 0.33 -0.60 -0.18 0.04 0.27 0.69 3000 1! s 3.04 0.01 0.37 2.41 2.78 3.02 3.27 3.80 2807 1! lp__ -443.66 0.18 9.50 -463.66 -449.84 -443.33 -437.14 -425.83 2805 1! ! Samples were drawn using NUTS(diag_e) at Tue Jun 17 15:45:43 2014.! For each parameter, n_eff is a crude measure of effective sample size,! and Rhat is the potential scale reduction factor on split chains (at ! convergence, Rhat=1).

> traceplot(fit)

OpenBUGS 1010.3 chain: 3 iteration: 10000 burn-in (warmup): 100 thin: 10 Mac Pro (2.8 GHz Quad-Core Intel Xeon) OS X 10.9.3 OpenBUGS 3.2.2 RStan/CmdStan 2.2.0

OpenBUGS OpenBUGS RStan CmdStan 0 75 150 225 300 compile run *OpenBUGSWine LinuxLinuxOpenBUGSStan

(2) Zero-Inated Negative Binomial Model

0 1 2 3 4 5 6 7 8 9 11 13 15 17 y count 0 10 20 30 40 0

: , : p(y| , , ) = (1 )+ NegBin(0| , ) (y = 0) NegBin(y| , ) (y > 0)

data {! int N;! int y[N];! }! parameters {! real theta;! real alpha;! real beta;! }! model {! // priors! theta ~ uniform(0, 1);! alpha ~ uniform(0, 100);! beta ~ uniform(0, 100);! for (i in 1:N) {! if (y[i] == 0) {! increment_log_prob(log((1 - theta)! + theta * (beta / (beta + 1))^alpha));! } else {! increment_log_prob(bernoulli_log(1, theta)! + neg_binomial_log(y[i], alpha, beta));! }! }! }! generated quantities {! real mu;! mu

Hamiltonian Monte Carloによるベイズ推定ソフトウェアStanの紹介

theta

fit

gt

int

incrementlogprob

Text of Hamiltonian Monte Carloによるベイズ推定ソフトウェアStanの紹介

Hamiltonian formulation and coherent structures in ... Hamiltonian formulation and coherent structures in electrostatic turbulence 2 1. Introduction The Hamiltonian formalism constitutes

Hamiltonian POV

Mekanika Hamiltonian

Hamiltonian Systems - Scholarpedia

Hamiltonian F - core.ac.uk

政治学方法論 I - 4. 統計分析の不確実性 - GitHub Pages ... ベイズ統計学? 統計分析の不確実性ベイズ統計学とベイズの定理ベイズの定理の使い方：よくある例

2．ビリーフとベイズの定理 ... 4月15日ベイズの定理とは？4月22日ベイズはどのようにして世に出たのか？5月6日【休日出勤】ベイズはコンピュータの父

Week 03 Hamiltonian

Hamiltonian Cycle

Polylog Hamiltonian simulation

Hamiltonian Graphs By: Matt Connor Fall 2013. Hamiltonian Graphs Abstract Algebra Graph Theory Hamiltonian Graphs

ベイズ･アプローチによる　　　　　　グラフィカル･テスト理論

Hamiltonian Cycles

―ベイズ推定とパーティクルフィルタ― swk/lecture/ic2010/kagami_ic... 知能制御システム学画像追跡(3) ―ベイズ推定とパーティクルフィルタ―

ベイズ Chow-Liu アルゴリズム

第1章確率に関するベイズの定理 - 221.114.158.246221.114.158.246/~bunken/statistics/ 第1章確率に関するベイズの定理基礎からのベイズ統計学輪読会

Quantum Hamiltonian Complexity

Hamiltonian Formalism

Hamiltonian Cycle (3A) · PDF file to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent

Hamiltonian Chaos 2

Hamiltonian Circuits

Hamiltonian Dynamics

[D2-28]ソフトウェア依存関係ネットワークにおけるベイズ的 pr.cei.uec.ac.jp/~terada/grad2019/poster/2017-master-watanabe- [d2-28]ソフトウェア依存関係ネットワークにおける

Non hamiltonian autonomous systems - Revista · PDF fileNon hamiltonian autonomous systems ... bngc ror aconsistent Hamiltonian ami Lagrangian formula- ... Togel lhe variable forthe

Hamiltonian Systems Ozorio

zhakarov hamiltonian

PRML 10.1節 ~ 10.3節 - 変分ベイズ法

Basic Hamiltonian mechanics

2011 Hamiltonian Catalog

Hamiltonian Graphs