A Blog Entry on Bayesian Computation by an Applied Mathematician
$$
$$
概要
(Gareth O. Roberts and Rosenthal, 2001) は (G. O. Roberts et al., 1997), (Gareth O. Roberts and Rosenthal, 1998) などの最適スケーリングの結果をまとめたサーベイ論文である．
1 導入
asymptotic overall acceptance rate が，MCMC アルゴリズムの効率性の指標に使える，という点を上手に導入している．
ひょっとしたら，(Chopin et al., 2022) が目指したい方向性は，同様の指標を SMC について見出すことだったのかもしれない．
Our results provide theoretical justification for a commonly used strategy for implementing the multivariate random-walk Metropolis algorithm. which dates back at least as far as (Tierney, 1994).
This demonstrates that MALA algorithms asymptotically mix considerably faster than do RWM algorithms.
2 乱歩 MH 法の提案分散
2.1 高次元極限
2.2 Markov 連鎖の効率性の査定
Chopin, N., Singh, S. S., Soto, T., and Vihola, M. (2022). On Resampling Schemes for Particle Filter with Weakly Informative Observations. The Annals of Statistics, 50(6), 3197–3222.
Roberts, G. O., Gelman, A., and Gilks, W. R. (1997). Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms. The Annals of Applied Probability, 7(1), 110–120.
Roberts, Gareth O., and Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to langevin diffusions. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 60(1), 255–268.
Roberts, Gareth O., and Rosenthal, J. S. (2001). Optimal Scaling for Various Metropolis-Hastings Algorithms. Statistical Science, 16(4), 351–367.
Tierney, L. (1994). Markov Chains for Exploring Posterior Distributions. The Annals of Statistics, 22(4), 1701–1728.