Result | trace plot 6 This chain seems good. Should I set initial values around (b, s_y)=(24, 0.3) ?
By the way 7 A contour map of log-posterior can be visualized. Because the number of parameters is just two in this model. For later explanation, I visualized 𝐸 (≈ − log-posterior ) 𝑇 𝐸: Energy, 𝑇: Temperature 𝐸 higher log-posterior lower 𝑇 previous model is correspond to 𝑇 = 1
Contour map of 𝐸/𝑇 (at 𝑇 = 1) 8
Contour map of 𝐸/𝑇 (at 𝑇 = 1) 9 around (24, 0.3) around (0.5, 0.3) 𝐸 ≈ 28 𝐸 ≈ 33
10 Setting initial values is generally difficult. Replica exchange MCMC (also known as parallel tempering)
References 11 • Y. Iba, "Extended Ensemble Monte Carlo," Int. J. Mod. Phys. C, 12, pp.623-652, 2001. • Movie by Y. Iba (in Japanese): レプリカ交換MCMC講義 （伊庭幸人） 難易度★★ https://www.youtube.com/watch?v=1c7mQIhEqmQ • Books (in Japanese) – 福島孝治 (2006) サイコロふって積分する方法 確率的情報処理と統 計力学(SGCライブラリ 50) p.60-66. – 伊庭ほか (2005) 計算統計II (統計科学のフロンティア 12) p.74-78
Overview 12 𝑁𝑟𝑒𝑝𝑙𝑖𝑐𝑎 = 5 𝑇5 > 𝑇4 > 𝑇3 > 𝑇2 > 𝑇1 = 1 Monte Carlo Step
Annealing and Heating 13 𝑇ℎ𝑖𝑔ℎ 𝑇𝑚𝑖𝑑𝑑𝑙𝑒 𝑇1 = 1
Sampling from joint distribution 14 • Theoretically, Replica exchange MCMC does sampling from the following joint distribution: 𝑁_𝑟𝑒𝑝𝑙𝑖𝑐𝑎 𝑝 𝜽1, … , 𝜽𝑇 =
Pseudo Code 16 set T of replicas: T = (1, T2, T3, ..., TN_replica) set initial values: x = (x0, x0, ..., x0) for e in 1, ..., N_exchange for r in 1, ..., N_replica short sampling started from x[r] at each T[r] end save MCMC samples at T = 1 exchange replicas if p < unif(0,1) for r in 1, ..., N_replica update x[r] end end
Discussion 23 • Can I re-use warmup result in Stan? i.e. step size & diagonal elements of inverse mass matrix • How to set iter and warmup of each short sampling? warmup=50 is usually too short as total sampling. But, total warmup is 50*100 in this case. Does it make sense?