Modeling Ices in star-forming clouds MCMC approac……

1、Modeling Ices in star-forming clouds:MCMC approachPavel Senin,following Jacqueline Keane presentationsAugust 11, 2006.Interstellar ices in star-forming regions:Image from P.Ehrenfreund & A.G.G.M.TielensIterative model Consists from equations that model physical processes that occur at the “surfa

2、ce” only or within the cloud only. Sequential iterations run considerably long period of time.MCMC Model Proven to be surprisingly effective in many cases. Surfaces, clouds and all processes an be combined in one model. Can be parallelized and tweaked to run “fast”. “Jump function” design demands fo

3、r significant amount of work.Bayesian Basics: Bayes theorem: (posterior = standardized likelihood * prior) Basically we need to specify the prior distribution (of molecules within ice clouds and at the “surface”)in order to apply Bayes theorem and get posterior distribution.(| ) ( )( |)(| ) ( )L Dpp

4、DL DpdMarkov Chains Basics: A Markov chain is a process that corresponds to the “no memory” network: To quantify the chain, we need to specify Initial probability: P(X1) Transition probability: P(Xt+1|Xt)X1X2X3Xn.What about Monte-Carlo? Monte-Carlo integration:babaniixpxfnxfEdxxpxfdxxh1)()(1)()()()(

5、| ) ( )( |)(| ) ( )L DppDL DpdRemember:MCMC Objectives: The basic idea is to set up a long chain which starts at a pre-specified distribution taken from sample (or randomly generated biased samples) then traverses to the posterior distribution that sampled for results. Distribution of Xn as n , is s

6、tationary distribution.Metropolis-Hastings algorithm Set an initial X1. If the chain is currently at Xn = x, randomly propose a new state Xn+1 = y according to a proposal density q(y | x). Accept the proposed jump with probability If not accepted, Xn+1 = x.)|()()|()(, 1minxyqxyxqy“Hacking” Metropoli

7、s-Hastings algorithm Our problem is multi-dimensional (various molecules, various transitions) can be parallelized by updating each component separately or by partitions using specified M-H schemes for each. A lot of tweaks already were found and published, maybe we will be able to apply some.Sampli

8、ng strategies “One thread” sampling: Simultaneous runs: Hybrid strategies?N chains = N samples+ one burn-in for allN samples from one chainburn-in periodConclusion: Assuming significant amount of work that already done for modeling physical processes “jump-function” design expected to be easier. Model tuning should be easier to due to significant amount of data existed. Bot