bring low. –Seneca, 1st century AD How, therefore, is she good, who without discernment comes to both the good and to the bad? It profits one nothing to worship her if she is truly fortune...let the bad worship her...this supposed deity. –St Augustine, 5th century AD
Understand the approach • Meet different strategies • Meet interface to MCMC, map2stan • See how to check chains for good health • See how to fix some common problems • Maybe start maximum entropy
5 6 7 p5 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. p4 (4) Move to proposal, with probability = p5 p4 proposal
5 6 7 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. (4) Move to proposal, with probability = p5 p4 (5) Repeat from (1)
5 6 7 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. (4) Move to proposal, with probability = p5 p4 (5) Repeat from (1) This procedure ensures visiting each island in proportion to its population, in the long run.
25 island number of weeks after 100 weeks 2 4 6 8 10 0 20 40 60 80 100 island number of weeks after 500 weeks 2 4 6 8 10 0 100 200 300 400 island number of weeks after 2000 weeks 0 2000 4000 6000 8000 10000 2 4 6 8 10 week island
the long run • No matter which island he starts on • Works as long as proposals are symmetric • Example of the Metropolis algorithm 2 4 6 8 10 0 500 1000 1500 island number of weeks after 10000 weeks
Monte Carlo (MCMC) • Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) into radicals (by collision, or possibly radiation, as in aromatic hydrocarbons). (b) The molecules dissociate, but the resulting radi- cals recombine without escaping from the liquid cage. (c) The molecules dissociate and escape from the cage. In this case we would not expect them to move more than a few molecular diameters through the dense medium before being thermalized. In accordance with the notation introduced by Burton, Magee, and Samuel,22 the molecules following 22 Burton, Magee, and Samuel, J. Chern. Phys. 20, 760 (1952). THE JOURNAL OF CHEMICAL PHYSICS tions that the ionized H2 0 molecules will become the H2 0t molecules, but this is not likely to be a complete correspondence. In conclusion we would like to emphasize that the qualitative result of this section is not critically de- pendent on the exact values of the physical parameters used. However, this treatment is classical, and a correct treatment must be wave mechanical; therefore the result of this section cannot be taken as an a priori theoretical prediction. The success of the radical diffu- sion model given above lends some plausibility to the occurrence of electron capture as described by this crude calculation. Further work is clearly needed. VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
Monte Carlo (MCMC) • Chain: Sequence of draws from distribution • Markov chain: History doesn’t matter, just where you are now • Monte Carlo: Random simulation Andrei Andreyevich Markov (Ма́рков) (1856–1922)
from a posterior distribution • “Islands”: parameter values • “Population size”: proportional to posterior probability • Works for any number of dimensions (parameters) • Works for continuous as well as discrete parameters
can’t write an integrated likelihood function • Many kinds of problems are like this: • Many multilevel models • Networks/phylogenies • Some spatial models
Models with many parameters usually have lots of highly correlated parameters • GS gets stuck, degenerates towards random walk • At best, inefficient because re-explores regions • Hamiltonian dynamics to the rescue • represent parameter state as particle in n- dimensional space • flick it around frictionless log-posterior • record positions William Rowan Hamilton (1805–1865) Commemorated on Irish Euro coin
of car: parameter vector • Speed of car: momentum of parameter values • Go fast when high • Go slow when low • Samples of position through time comprise samples from posterior distribution
−3 −2 −1 0 1 2 3 iteration last position coordinate Random−walk Metropolis 0 200 400 600 800 1000 −3 −2 −1 0 1 2 3 iteration last position coordinate Hamiltonian Monte Carlo gure 6: Values for the variable with largest standard deviation for the 100-dimension ample, from a random-walk Metropolis run and an HMC run with L = 150. To mat mputation time, 150 updates were counted as one iteration for random-walk Metropoli te works only when, as here, the distribution is tightly constrained in only one directio Neal. 2011. MCMC from Hamiltonian dynamics
Augustin-Louis Cauchy (KO-shee) • Ratio of two Gaussian samples • Useful distribution with thick tails • Parameters: location of mode, scale • Mean and variance undefined • Related to Lévy flights Baron Augustin-Louis Cauchy (1789–1857)
rejected because of the following issue: Error in function stan::prob::multi_normal_log(N4stan5 Covariance matrixCovariance matrix is not positive definite. Covariance matrix(0,0) is 0:0. If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine, but if this warning occurs often then your model may be either severely ill-conditioned or misspecified. Don’t freak out, when you see this: If it happens some during warmup (adaptation), that’s normal. If it happens a lot, after warmup, then freak out.
efficient sampling • Samples during warmup NOT from posterior • Automatically discarded by precis/summary and other functions • Warmup is NOT “burn in” — burn in is front part of Markov chain using Gibbs or Metropolis &"4: ).$ warmup
number of samples • First run: Just getting comfortable • warmup=1000, iter=2000 fine • Eventually: • Enough warmup so mixing good, sampling efficient • Enough sampling for your purpose (n_eff) • Means? 200 usually enough • 99th centile? 20-thousand or more • Poorly mixing chains always require more samples
chains • First run: One chain, see how it goes • Tune: Multiple chains, to check convergence/ stationarity • Final run: One chain is fine, more if you have processors to spare • My heuristic: 1 short / 4 medium / 2 long
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