Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.386 29 — Monte Carlo Methodswhen B proposals are concatenated will be the end-point of a random walkwith B steps in it. This walk might have mean zero, or it might have atendency to drift upwards (if most moves increase the energy and only a fewdecrease it). In general the latter will hold, if the acceptance rate f is small:the mean change in energy from any one move will be some ∆E > 0 and sothe acceptance probability for the concatenation of B moves will be of order1/(1 + exp(−B∆E)), which scales roughly as f B . The mean-square-distancemoved will be of order f B Bɛ 2 , where ɛ is the typical step size. In contrast,the mean-square-distance moved when the moves are considered individuallywill be of order fBɛ 2 .1.110.90.80.70.60.50.40.3100010000100000theory1086420.2000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6100010000100000theory3.532.521.510.5Solution to exercise 29.13 (p.382). The weights are w = P (x)/Q(x) and x isdrawn from Q. The mean weight is∫∫dx Q(x) [P (x)/Q(x)] = dx P (x) = 1, (29.57)assuming the integral converges. The variance is∫ [ ] P (x) 2var(w) = dx Q(x)Q(x) − 1 (29.58)∫= dx P (x)2 − 2P (x) + Q(x) (29.59)Q(x)[∫= dx Z (QZP2 exp(− x2 22 σp2 − 1 ))]σq2 − 1, (29.60)Figure 29.20. Importancesampling in one dimension. ForR = 1000, 10 4 , and 10 5 , thenormalizing constant of aGaussian distribution (known infact to be 1) was estimated usingimportance sampling with asampler density of standarddeviation σ q (horizontal axis).The same random number seedwas used for all runs. The threeplots show (a) the estimatednormalizing constant; (b) theempirical standard deviation ofthe R weights; (c) 30 of theweights.where Z Q /ZP 2 = σ q/( √ 2πσp 2 ). The integral in (29.60) is finite only if thecoefficient of x 2 in the exponent is positive, i.e., ifIf this condition is satisfied, the variance isvar(w) =σ 2 q > 1 2 σ2 p . (29.61)σ (q√ 2√ 2π2πσ 2 pσp2 − 1 ) −12 σq2 − 1 =σq2 ( )σ p 2σ 2 q − σp2 1/2− 1. (29.62)As σ q approaches the critical value – about 0.7σ p – the variance becomesinfinite. Figure 29.20 illustrates these phenomena for σ p = 1 with σ q varyingfrom 0.1 to 1.5. The same random number seed was used for all runs, sothe weights and estimates follow smooth curves. Notice that the empiricalstandard deviation of the R weights can look quite small and well-behaved(say, at σ q ≃ 0.3) when the true standard deviation is nevertheless infinite.