Gibbs sampling
From Wikipedia, the free encyclopedia Jump to:navigation, search In mathematics and physics, Gibbs sampling or Gibbs sampler is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables. The purpose of such a sequence is to approximate the joint distribution, or to compute an integral (such as an expected value). Gibbs sampling is a special case of the Metropolis–Hastings algorithm, and thus an example of a Markov chain Monte Carlo algorithm. The algorithm is named after the physicist J. W. Gibbs, in reference to an analogy between the sampling algorithm and statistical physics. The algorithm was devised by brothers Stuart and Donald Geman, some eight decades after the passing of Gibbs.[1] Gibbs sampling is applicable when the joint distribution is not known explicitly, but the conditional distribution of each variable is known. The Gibbs sampling algorithm generates an instance from the distribution of each variable in turn, conditional on the current values of the other variables. It can be shown (see, for example, Gelman et al. 1995) that the sequence of samples constitutes a Markov chain, and the stationary distribution of that Markov chain is just the sought-after joint distribution. Gibbs sampling is particularly well-adapted to sampling the posterior distribution of a Bayesian network, since Bayesian networks are typically specified as a collection of conditional distributions. Contents[hide] 1 Background 2 Implementation 3 Failure modes 4 Software 5 Notes 6 References 7 External links // [edit] Background Gibbs sampling is a special case of Metropolis–Hastings algorithm. The point of Gibbs sampling is that given a multivariate distribution it is simpler to sample from a conditional distribution than to marginalize by integrating over a joint distribution. Suppose we want to obtain samples of from a joint distribution . We begin with a value of and sample by . Once that value of is calculated, repeat by sampling for the next : . [edit] Implementation Suppose that a sample is taken from a distribution depending on a parameter vector of length , with prior distribution . It may be that is very large and that numerical integration to find the marginal densities of the would be computationally expensive. Then an alternative method of calculating the marginal densities is to create a Markov chain on the space by repeating these two steps: Pick a random index Pick a new value for according to These steps define a reversible Markov chain with the desired invariant distribution . This can be proved as follows. Define if for all and let denote the probability of a jump from to . Then, the transition probabilities are So since is an equivalence relation. Thus the detailed balance equations are satisfied, implying the chain is reversible and it has invariant distribution . In practice, the suffix is not chosen at random, and the chain cycles through the suffixes in order. In general this gives a non-reversible chain, but it will still have the desired invariant distribution (as long as the chain can access all states under the fixed ordering). [edit] Failure modes There are two ways that Gibbs sampling can fail. The first is when there are islands of high-probability states, with no paths between them. For example, consider a probability distribution over 2-bit vectors, where the vectors (0,0) and (1,1) each have probability ½, but the other two vectors (0,1) and (1,0) have probability zero. Gibbs sampling will become trapped in one of the two high-probability vectors, and will never reach the other one. More generally, for any distribution over high-dimensional, real-valued vectors, if two particular elements of the vector are perfectly correlated (or perfectly anti-correlated), those two elements will become stuck, and Gibbs sampling will never be able to change them. The second problem can happen even when all states have nonzero probability and there is only a single island of high-probability states. For example, consider a probability distribution over 100-bit vectors, where the all-zeros vector occurs with probability ½, and all other vectors are equally probable, and so have a probability of each. If you want to estimate the probability of the zero vector, it would be sufficient to take 100 or 1000 samples from the true distribution. That would very likely give an answer very close to ½. But you would probably have to take more than 2100 samples from Gibbs sampling to get the same result. No computer could do this in a lifetime. This problem occurs no matter how long the burn in period is. This is because in the true distribution, the zero vector occurs half the time, and those occurrences are randomly mixed in with the nonzero vectors. Even a small sample will see both zero and nonzero vectors. But Gibbs sampling will alternate between returning only the zero vector for long periods (about 299 in a row), then only nonzero vectors for long periods (about 299 in a row). Thus convergence to the true distribution is extremely slow, requiring much more than 299 steps; taking this many steps is not computationally feasible in a reasonable time period. The slow convergence here can be seen as a consequence of the curse of dimensionality. [edit] Software The WinBUGS software (the open source version is called OpenBUGS) does a Bayesian analysis of complex statistical models using Markov chain Monte Carlo. BUGS comes from Bayesian inference using Gibbs sampling. JAGS (Just another Gibbs sampler) is a GPL program for analysis of Bayesian hierarchical models using Markov Chain Monte Carlo. [edit] Notes --- On Fri, 5/14/10, Sole Acha, Xavi <[email protected]> wrote: From: Sole Acha, Xavi <[email protected]> Subject: Re: [BiO BB] How to create a Transcription binding site profile To: "General Forum at Bioinformatics.Org" <[email protected]> Date: Friday, May 14, 2010, 11:33 AM Clustal may work for you. http://www.clustal.org/ HTH, Xavi. ------ Xavier Solé Acha Unitat de Biomarcadors i Susceptibilitat Unit of Biomarkers and Susceptibility Institut Català d'Oncologia // Catalan Institute of Oncology Gran Via de L'Hospitalet 199-203 08907 L'Hospitalet de Llobregat, Barcelona, Spain. Phone: +34 93 260 71 86 / +34 93 335 90 11 (ext. 3194) Fax: +34 93 260 71 88 E-mail: x.sole (at) iconcologia.net -----Mensaje original----- De: [email protected] [mailto:[email protected]] En nombre de Dan Bolser Enviado el: viernes, 14 de mayo de 2010 17:30 Para: General Forum at Bioinformatics.Org Asunto: Re: [BiO BB] How to create a Transcription binding site profile I don't know anything specific, but you could try using 'Gibbs sampling' algorithms to automatically discover the motif in the sequence sets? On 13 May 2010 21:02, Che, Anney (NIH/NCI) [E] <[email protected]> wrote: > > Hi everyone, > > I have questions regarding on how to create a transcription binding sites > profile. > > Since I have the sequences of the areas that bind to the gene from Chip-ChIp > , I can align all the sequences then with the alignment that created to > generate a profile. > > Since this is high-throughput, does any one know any tool that can align > short sequences programmatically and then output an alignment file? > > Also a program that generates an alignment profile from an alignment. > > > Thanks, > > Anney > > _______________________________________________ > BBB mailing list > [email protected] > http://www.bioinformatics.org/mailman/listinfo/bbb > _______________________________________________ BBB mailing list [email protected] http://www.bioinformatics.org/mailman/listinfo/bbb _______________________________________________ BBB mailing list [email protected] http://www.bioinformatics.org/mailman/listinfo/bbb _______________________________________________ BBB mailing list [email protected] http://www.bioinformatics.org/mailman/listinfo/bbb
