Metropolis Hastings Algorithm
As already pointed out, the main purpose of this paper is to calculate the tail of the distribution of optimum scores of gapped local alignments over pairs of randomly and independently drawn sequences of finite lengths. The basic idea of our approach is to generate the sequences from different distributions, which are biased towards higher scores.
In order to be more precise let us denote the state space of all possible pairs of sequences (x, y) as X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@ and an element in this space as a configuration. We write X = (x, y).
The probability mass function (pmf) of finding X under the null model is given by p(X)=p(x,y)=∏i=1Lfxi∏j=1Mfyj
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakieqacqWFybawcqGGPaqkcqGH9aqpcqWGWbaCcqGGOaakcqWF4baEcqGGSaalcqWF5bqEcqGGPaqkcqGH9aqpdaqeWaqaaiabdAgaMnaaBaaaleaacqWG4baEdaWgaaadbaGaemyAaKgabeaaaSqabaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemitaWeaniabg+GivdGcdaqeWaqaaiabdAgaMnaaBaaaleaacqWG5bqEdaWgaaadbaGaemOAaOgabeaaaSqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemyta0eaniabg+Givdaaaa@4FD5@ and the alignment score as defined in Eq. (2) is a random variable. A direct way to obtain the probability of the occurrence of a certain score s, is to generate n uncorrelated representatives Xi ∈ X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@ according to the null model and then compute the expectation values of the family of indicator functions hs: X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@ → ℝ with hs (X) = 1, if S (X) = s and hs (X) = 0 otherwise, in other words
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=LriqjabcUfaBjabdofatjabcIcaOGqabiab+HfayjabcMcaPiabg2da9iabdohaZjabc2faDjabg2da9iab=ri8fjabcUfaBjabdIgaOnaaBaaaleaacqWGZbWCaeqaaOGaeiikaGIae4hwaGLaeiykaKIaeiyxa0Laeyypa0ZaaabuaeaacqWGObaAdaWgaaWcbaGaem4CamhabeaaaeaacqGFybawaeqaniabggHiLdGccqGGOaakcqGFybawcqGGPaqkcqWGWbaCcqGGOaakcqGFybawcqGGPaqkcqGHijYUdaWcaaqaaiabigdaXaqaaiabd6gaUbaadaaeWbqaaiabdIgaOnaaBaaaleaacqWGZbWCaeqaaOGaeiikaGIae4hwaG1aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aOGaeiOla4caaa@6E0F@
Since the region of biological interest is located in the rare-event tail a huge amount of samples would be needed to achieve an acceptable accuracy. In practice the rare-event tail becomes inaccessible.
Our method is based on importance sampling of a mixture of chains based on the Metropolis-Hastings algorithm. Before describing the coupling of multiple chains, we introduce the general idea of importance sampling first: The approach is based on sampling from a different distribution, such that the region of interest is sampled with high probability. Since this happens in a controlled manner the true distribution can be obtained afterward, as frequently used in variance reduction techniques. The modified distribution yields a different random variable with a different pmf q. We may write
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqGGOaakcqWGZbWCcqGGPaqkcqGH9aqptuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=LriqjabcUfaBjabdofatjabcIcaOGqabiab+HfayjabcMcaPiabg2da9iabdohaZjabc2faDjabg2da9maaqafabaGaemiAaG2aaSbaaSqaaiabdohaZbqabaGccqGGOaakcuGFybawgaqbaiabcMcaPmaalaaabaGaemiCaaNaeiikaGIaf4hwaGLbauaacqGGPaqkaeaacqWGXbqCcqGGOaakcuGFybawgaqbaiabcMcaPaaaaSqaaiqb+HfayzaafaaabeqdcqGHris5aOGaemyCaeNaeiikaGIaf4hwaGLbauaacqGGPaqkcqGHijYUdaWcaaqaaiabigdaXaqaaiabd6gaUbaadaaeWbqaaiabdIgaOnaaBaaaleaacqWGZbWCaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aOGaeiikaGIaf4hwaGLbauaadaWgaaWcbaGaemyAaKgabeaakiabcMcaPmaalaaabaGaemiCaaNaeiikaGIaf4hwaGLbauaadaWgaaWcbaGaemyAaKgabeaakiabcMcaPaqaaiabdghaXjabcIcaOiqb+HfayzaafaWaaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaaGaeiOla4caaa@7C9B@
At least approximately, the distribution of local alignment follows a Gumbel distribution, which exhibits an exponential behavior in the tail. Therefore an obvious choice for the biased distribution is
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaWgaaWcbaGaemivaqfabeaakiabcIcaOGqabiab=HfayjabcMcaPiabggMi6oaalaaabaGafmyCaeNbaGaadaWgaaWcbaGaemivaqfabeaakiabcIcaOiabdIfayjabcMcaPaqaaiabdQfaAnaaBaaaleaacqWGubavaeqaaaaakiabggMi6oaalaaabaGaeGymaedabaGaemOwaO1aaSbaaSqaaiabdsfaubqabaaaaOGaemiCaaNaeiikaGIae8hwaGLaeiykaKIaeyyXICTagiyzauMaeiiEaGNaeiiCaaNaei4waSLaem4uamLaeiikaGIae8hwaGLaeiykaKIaei4la8IaemivaqLaeiyxa0LaeiilaWcaaa@5680@
where q˜T
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGXbqCgaacamaaBaaaleaacqWGubavaeqaaaaa@2F83@ the unnormalized weight of a configuration, ZT is a (usually unknown) normalization constant and T an adjustable parameter, which we will call "temperature" (In the framework of statistical mechanics, which is closely related to our method, the parameter T describes the temperature of a physical system. The pair of sequences can be seen as a configuration of a physical system and the negative score as the energy function. Then exp [S (X)/T] refers to the so called Gibbs-Boltzmann distribution.) The close-to Gumbel form of the distribution is also directly related to the so called "large deviation rate function", which basically describes the decay rate of the tail of the distribution. Note that, if the score distribution is an exact Gumbel distribution Eq. (3), i.e. the rate function a known constant λ, then setting T = 1/λ in Eq. (7) yields a "flat score histogram" for sufficient large s. Hence, in this case, a simulation at a single carfully chosen value T would be sufficient to obtain the full result. Since P (s) does not follow the Gumbel form exactly, importance sampling has to be applied. Each value of T selects one region of the distribution around which a high accurracy is obtained.
This importance sampling approach is conceptual related to the method of "measure change" in large deviation theory. For example Siegmund and Yakir 14 approximated the p-value for local sequence alignment by considering the log-likelihood ratio between an alternative measure and the measure of the null model. Under the new measure a rare event occurs more likely than under the original null measure and approximations become possible. Another example can be found in Ref. 29, where techniques from large deviation theory were applied to proof "asymptotic efficiency" of rare-event simulations.
However, since there is no direct method to sample directly according to the modified distribution Eq. (7) we implemented the Metropolis-Hastings algorithm 30, which is explained now in detail. It is based on ergodic Markov chain Monte Carlo (MCMC) in state space. Ergodic here means, that for a given state in the configuration space X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@ any other can be achieved by stepwise "local" modifications of configurations in finite time. Note that we work in discrete time steps here. Let X ∈ X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@ a configuration at time t (e.g. at the start of the simulation). To determine the configuration at time t + 1, first a trial configuration X* is selected randomly among its "neighbors". The neighborhood of a configuration depends on the choice of trial steps, which are specified below. For practical reasons we require, that the score within a neighborhood of a given configuration will not change too much. The transition matrix for this trial selection process is denoted by P (X, X*). Now, the trial configuration becomes the configuration at time t + 1, i.e. is accepted, with probability
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaacaiabcIcaOGqabiab=HfayjabgkziUkab=HfaynaaCaaaleqabaGaey4fIOcaaOGaeiykaKIaeyypa0JagiyBa0MaeiyyaeMaeiiEaG3aaiWabeaacqaIXaqmcqGGSaaldaWcaaqaaiabdcfaqjabcIcaOiab=HfaynaaCaaaleqabaGaey4fIOcaaOGaeiilaWIae8hwaGLaeiykaKcabaGaemiuaaLaeiikaGIae8hwaGLaeiilaWIae8hwaG1aaWbaaSqabeaacqGHxiIkaaGccqGGPaqkaaGaeyyXIC9aaSaaaeaacqWGXbqCdaWgaaWcbaGaemivaqfabeaakiabcIcaOiab=HfaynaaCaaaleqabaGaey4fIOcaaOGaeiykaKcabaGaemyCae3aaSbaaSqaaiabdsfaubqabaGccqGGOaakcqWFybawcqGGPaqkaaaacaGL7bGaayzFaaGaeyypa0JagiyBa0MaeiyyaeMaeiiEaG3aaiWabeaacqaIXaqmcqGGSaaldaWcaaqaaiabdcfaqjabcIcaOiab=HfaynaaCaaaleqabaGaey4fIOcaaOGaeiilaWIae8hwaGLaeiykaKcabaGaemiuaaLaeiikaGIae8hwaGLaeiilaWIae8hwaG1aaWbaaSqabeaacqGHxiIkaaGccqGGPaqkaaGagiyzauMaeiiEaGNaeiiCaaNaei4waSLaeuiLdqKaem4uamLaei4la8IaemivaqLaeiyxa0facaGL7bGaayzFaaGaeiilaWcaaa@8037@
with ΔS = S (X*) - S (X) If the trial configuration is not accepted, the previous configuration X is kept for the next time step t + 1. In this way, the Markov chain fulfills the detailed balance condition P (X*, X)p˜
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaacaaaa@2E24@(X* → X)·qT (X*) = P (X, X*)p˜
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaacaaaa@2E24@(X → X*)·qT(X). In this case it has been proven that an ergodic Markov chain converges to the stationary distribution qT. Ergodicity means, that there is a non-zero probability for a path between any pair(X1, X2) of configurations.
We used a simple way to define the neighborhood of a configuration and constructed the trial configuration as follows: First a letter a is drawn from the alphabet Σ according to the letter weights fa and next one of the sequences (x or y) and a position i is chosen randomly. Finally, the letter at position i is replaced by a.
Given a Monte Carlo chain (X1, ..., Xn) estimated for a fixed temperature T in principle one may estimate expectation values with respect to any member of the family of distributions qT by importance reweighting
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Since the normalization of qT is not trivial, we used a different normalization
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=ri8fnaaBaaaleaacuWGubavgaqbaaqabaGccqGGBbWwcqWGNbWzcqGGOaakieqacqGFybawcqGGPaqkcqGGDbqxcqGHijYUdaWcaaqaaiabigdaXaqaaiabd6gaUbaadaaeWbqaamaalaaabaGafmyCaeNbaGaadaWgaaWcbaGafmivaqLbauaaaeqaaOGaeiikaGIae4hwaG1aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaeaacuWGXbqCgaacamaaBaaaleaacqWGubavaeqaaOGaeiikaGIae4hwaG1aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaaaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoakiabgwSixlabdEgaNjabcIcaOiab+HfaynaaBaaaleaacqWGPbqAaeqaaOGaeiykaKIaeiilaWcaaa@63A2@
and estimate Z from the sample Z=∑k=1nq˜T′(Xk)/q˜T(Xk)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGAbGwcqGH9aqpdaaeWaqaaiqbdghaXzaaiaWaaSbaaSqaaiqbdsfauzaafaaabeaaaeaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoakiabcIcaOGqabiab=HfaynaaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaei4la8IafmyCaeNbaGaadaWgaaWcbaGaemivaqfabeaakiabcIcaOiab=HfaynaaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaaa@4556@. A detailed discussion about this issue can be found in Ref. 3132. In practice this may work badly as soon as the parameter ranges of the given distribution and the target distribution do not overlap sufficiently. In this case qT'(Xi) is very small, but the configurations where qT' (X)/qT (X) is sufficiently large are not generated because qT (X) is relatively small for those. Therefore we sampled a mixture of many coupled Monte Carlo chains and reweighted the mixture, which is explained in detail in the next section. This allows for large overlap between neighboring distributions and to determine the normalization constants, up to an irrelevant global constant.
Metropolis Coupled MCMC
Metropolis Coupled Markov Chain Monte Carlo (MCMCMC) was first invented by Charles Geyer 33 and then reinvented by Hukushima and Nemoto 34 under the term exchange Monte Carlo. In physical literature MCMCMC is often denoted as parallel tempering. The method has become a standard tool in disordered systems with a rough (free) energy landscape 35. These rough energy landscapes are characterized by high energy barriers and can be found for problems like protein folding 3637383940, nucleation 41, spin-glasses 4243 and other models characterized by rare events 1944. In the last decade it turned out that MCMCMC accelerates equilibration and mixing remarkably.
In the framework of MCMCMC m copies X(1), ..., X(m) of the system held at different temperatures T1 <T2 < ... <Tm are simulated in parallel. This means one samples from the product of the state space X
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFxepwaaa@384F@m weighted with the joint distribution with weights ∏j=1mqTj
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqeWaqaaiabdghaXnaaBaaaleaacqWGubavdaWgaaadbaGaemOAaOgabeaaaSqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemyBa0ganiabg+Givdaaaa@37A5@. Since the different copies are allowed to exchange temperatures during the simulation, let us define the space of all possible mappings from the m configurations to the m temperatures as temperature space.
During the simulation, mainly each of the replicated configurations will evolve independently according the underlying MCMC scheme charaterized by the weight Eq. (7) at its current temperature, i.e. according to Eq. (8). In addition to this evolution, every texchangeth step (for each replicated configuration) a flip between two neighboring replicas k and k + 1 is attempted, i.e. for all k ∈ {1, ..., m - 1}. If an attempt is successful, the configurations X(k) and X(k+1) are exchanged (denoted by X(k) ↔ X(k+1)), i.e. the configurations which has previously evolved at temperature Tk will now evolve at temperature Tk + 1 and vice versa. This exchange is accepted with the probability
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaGafmiCaaNbaGaadaqadaqaaGqabiab=HfaynaaCaaaleqabaGaeiikaGIaem4AaSMaeiykaKcaaOGaeyiLHSQae8hwaG1aaWbaaSqabeaacqGGOaakcqWGRbWAcqGHRaWkcqaIXaqmcqGGPaqkaaaakiaawIcacaGLPaaacqGH9aqpcyGGTbqBcqGGHbqycqGG4baEdaGadeqaaiabigdaXiabcYcaSmaalaaabaGaemyCae3aaSbaaSqaaiabdsfaunaaBaaameaacqWGRbWAaeqaaaWcbeaakiabcIcaOiab=HfaynaaCaaaleqabaGaeiikaGIaem4AaSMaey4kaSIaeGymaeJaeiykaKcaaOGaeiykaKcabaGaemyCae3aaSbaaSqaaiabdsfaunaaBaaameaacqWGRbWAaeqaaaWcbeaakiabcIcaOiab=HfaynaaCaaaleqabaGaeiikaGIaem4AaSMaeiykaKcaaOGaeiykaKcaaiabgwSixpaalaaabaGaemyCae3aaSbaaSqaaiabdsfaunaaBaaameaacqWGRbWAcqGHRaWkcqaIXaqmaeqaaaWcbeaakiabcIcaOiab=HfaynaaCaaaleqabaGaeiikaGIaem4AaSMaeiykaKcaaOGaeiykaKcabaGaemyCae3aaSbaaSqaaiabdsfaunaaBaaameaacqWGRbWAcqGHRaWkcqaIXaqmaeqaaaWcbeaakiabcIcaOiab=HfaynaaCaaaleqabaGaeiikaGIaem4AaSMaey4kaSIaeGymaeJaeiykaKcaaOGaeiykaKcaaaGaay5Eaiaaw2haaaqaaiabg2da9iGbc2gaTjabcggaHjabcIha4naacmqabaGaeGymaeJaeiilaWIagiyzauMaeiiEaGNaeiiCaaNaei4waSLaeyOeI0IaeuiLdqecciGae4NSdi2aaSbaaSqaaiabdUgaRbqabaGccqqHuoarcqWGtbWucqGGDbqxaiaawUhacaGL9baacqGGSaalaaaaaa@92A4@
where, Δβk=1Tk+1−1Tk
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoariiGacqWFYoGydaWgaaWcbaGaem4AaSgabeaakiabg2da9maalaaabaGaeGymaedabaGaemivaq1aaSbaaSqaaiabdUgaRjabgUcaRiabigdaXaqabaaaaOGaeyOeI0YaaSaaaeaacqaIXaqmaeaacqWGubavdaWgaaWcbaGaem4AaSgabeaaaaaaaa@3C96@, ΔS = S (X(k + 1)) - S (Xk) and all weights are calculated with the configurations before the flip. This leads to a "random walk in temperature space" of the configurations.
Note that another possible approach based on Markov chains to compute p-values of a random model with a random variable X, ℙ [X > b] was introduced by Wilbur 45. The first step is to sample from an unbiased Markov chain based on the model of interest and compute the median of the (high probability) distribution. In the second iteration the random walk is truncated such that only values larger than the median of the first iteration occur. This corresponds to choosing a lower temperaure T in Eq. (7). The third iteration uses the median of the second iteration and so forth. This is repeated until a fraction of 1/4 of all events lay beyond a certain threshold value leading to a non decreasing sequence of splitting intervals defined by the medians of each iteration. This sequence is used in the second stage of the algorithm, where p-values are computed explicitly by multiplying the p-values of the truncated distribution in each iteration.
Although this method is easy to implement and errors can be estimated relatively simply, the MCMCMC approach has the advantage that the different configurations are not subjected to a sequence of decreasing temperatures, but perform a random walk in temperature space, i.e. visit all temperatures several times. Thus, mixing is accelerated and hence fewer Monte Carlo steps are required.