Surveying the Characteristics of Population Monte Carlo

Transcription

1 International Research Journal of Applied and Basic Sciences 2013 Available online at ISSN X / Vol, 7 (9): Science Explorer Publications Surveying the Characteristics of Population Monte Carlo Ehsan Fayyazi 1*, Gholamhossein Gholami 2 1. Department of Statistics, Science and Research Branch, Islamic Azad University, Fars, Iran. 2. Department of Mathematics, Faculty of sciences, Urmia University, Urmia, Iran. Corresponding Author e.fayyazi@fsriau.ac.ir ABSTRACT: The importance sampling method as other Monte Carlo Markov Chain (MCMC) algorithms is iterative while this algorithm i.e. importance sampling does not depend on the initiating point. The Population Monte Carlo method includes the frequent production of importance sampling whose used importance functions depend on the previous produced importance samples. The advantage of this method over the MCMC algorithm is that the framework of this algorithm in each iteration is unbiased, so running this algorithm can stop at any given time. The reason is that the iterations improve running the importance function (i.e. the proposal distribution). Hence, this leads to the improved importance sampling. In this study, we survey this method through diverse examples. Keywords: Population Monte Carlo, Importance sampling, Monte Carlo Markov Chain (MCMC), mixed models, Metropolis-Hastings algorithms. INTRODUCTION This study suggests a method named Population Monte Carlo (PMC) which is the combination of Monte Carlo Chain methods, importance sampling, and importance resampling. The method takes the advantages of every of these methods. In doing so, we describe the extension of importance sampling, andthen suggest the population Monte Carlo. Population Monte Carlo Population Monte Carlo (PMC) algorithm is an iterative importance sampling method which produces in each iteration the stimulated approximate sample from the target distribution and the adaptive algorithm which arranges the proposal distribution with the target distribution throughout the iterations. So, the theoretical basis of this method rootin the importance sampling rather than in MCMC and despite the iterated characteristics (that is, unbiased at least to the order O(1/n)), the estimation of target distribution is valid in each iteration and does not require the convergence times and stopping rules. simulating the sample Considering the MCMC, stationary distribution has been taken into account as a limit distribution from Markovsequenceas{ }, having this experimental result is large enough for X to t. A rather simple expansion of this perspective is that instead of simulating a distribution point of π we embark on simulating the n number sample distributed from π. In other words, one would simulate n number from the following π (x,, x ) = π(x ) The expansion in [4] and [5] accompanied by developed programming for nmcmc parallel running has been argued. In fact, one would use this complete sample int iteration for devising a proposal distribution in + 1 iteration. General importance sampling The PMC algorithm can be considered in more general framework: one would assume differentproposal distributions in each iteration and for each particle in this algorithm. In other words, if i is the sample indices and t the iteration indices, then X ( ) can be simulated from q distributions which might depend on the previous samples while being independent from other samples (to be conditioned on other samples) X ( ) ~q (x)

2 So, an important weight is assumed to any simulated point Thus, the following estimations ϱ ( ) = π(x( ) ) q (x ( ) ) I = 1 n ϱ ( ) h(x ( ) ) are the asymmetric estimators [h(x)][1]. The reason that the importance distributions of q can depend on the all previous experiments without changingϱ ( ) is unbiasedness. In fact, we have: E ϱ ( ) h x ( ) = π(x) h(x)q (x)dxg(ς)dς q (x) = h(x)π(x)dxg(ς)dς = [h(x)] (1) in which ς is the preceding random variations vector existed in q and g(ς) is its arbitrary distribution ( q are independent of g(ς)) [1]. In addition, depending on previous status does not contribute to the correlation; since ϱ ( ) h x ( ) ϱ ( ) h x ( ) π(x) π(y) = h(x)q (x)dx h(y)q (y)dyg(ς)dς q (x) q (y) = h(x)π(x)dx h(y)π(y)dyg(ς)dς = [h(x)] (2) so that ς is the previous random variations existing either in q or q and g is its density. Also, assuming that the following variances Var(ϱ ( ) h x ( ) ) areexist for every 1 i n, which means that the proposal distributions of q compared to π, we have: should be heavier tailed Var(I ) = 1 n Var(ϱ ( ) h x ( ) ) (3) Because the effects of ϱ ( ) weights are declined. In fact, even if the x ( ) s are correlated, the weighted terms will be not correlated according to the following theorem [1]. Theorem 1.1. Suppose{X } is a Markov chain with transition kernel q, then: π(x ) Var( h(x ) q(x X ) ) = Var(h(X ) π(x ) q(x X ) ). Proof: without lossof generality, suppose [h(x)] = 0. If we define: ω = π(x ) q(x X ), then, the covariance between ω h(x ) and ω h(x ) equals as: [ω h(x )ω h(x )] = [ (ω h(x X ) ω h(x X ) ] in which we iterated the expectation and used Markov s conditional independency attribute. The second conditional expectation is as follows: π(z) (ω h(x ) X ) = h(z) q(z X )dz q(z X ) = [h(x)] = 0, Which shows that covariance equals to zero [3]. So, the sum of variance equals to the sum of each of the terms for the importance sampling estimators. Note that resampling can be included in some iteration or even in all of the algorithms iterations, but in that case there will be no increase in weights throughout the iterations compared to the particle systems. Similar to most of the cases, the target distribution π lacks any scale and instead we use the following scale 523

3 ϱ ( ) π(x( ) ) q (x ( ) ), i = 1,, n whose sum of ϱ ( ) weights equals to 1. In this case, both the above unbiasedness and variance analysis are declined, although they still continue [3]. In fact, each constant normalizing estimator of π is improved by t iteration since it is the total average of convergent estimator from the normalizing inverse constant [3]. ω = 1 tn π(z ( ) ) q (z ( ) ) So, when t increases, ω participates less in symmetric status and I variations. The above attribute can be considered for t as large enough. Moreover, if ω in equation (4) is used for ω, i.e. if ϱ ( ) = π(z( ) ) ω q ( ) ( ) then, the variance analysis (3) is recoverable by means of analyzing the same conditions [3]. Population Monte Carlo Suppose, we are interested in estimating the following integral I = h(x) π(x)dx We tend to sample π(x) from the target distribution. The Population Monte Carlo algorithm provides us with a sample population from the targetdistribution to be generated in each iteration. The reason for naming this method as population Monte Carlo is the indication on simulation idea of one complete sample as iterated, rather than simulating iteratively the approximate samples points. Since the above section confirms that one iterated importance sampling method based on the dependent sample s proposal distributionis fundamentally one specific importance sampling, one would proposal algorithm 1 which is confirmed by importance sampling principles. Algorithm 1 Population Monte Carlo (4) In this algorithm, the indication is on level (i) since it is the main attribute of PMC algorithm. The proposal distributions can be considered separately in each level without declining the validity of method. So, the proposal distributions of q can be chosen according to the previous proposal distributions of q ( ) and in particular, they can depend on the previous sample (x ( ),, x ( ) ) or even on all of the simulated samples. In each t iteration, the PMC estimator is as follows: I = 1 n ϱ ( ) h(x ( ) ) Unbiasedness of this estimator was shown in (1). Although, in many cases the π(x) distribution can probably be identifiable to the extent of normalizing constant. Like importance sampling, one can use the following alternative estimator: I = ϱ( ) h(x ( ) ) ϱ ( ) In this cane, the unbiasedness of estimator is removed; however, it is taken into account as consistent estimator (Cappé et al., 2004). 524

4 In practice, one would calculate the man of the whole iterations in order to improve estimation. A PMC cumulative estimator on all iterations is defined as follows: I = ( ( ) ( ) ( ) ( ) ) where is the weights to mix the estimators of different iterations. Efficient chooses of which minimize estimator variance of 5 is as follows (Douc et al., 2005): = ( ) where is the estimator variance of I in iteration. Note that it might be possible for the importance weights tobe symmetrical; this symmetrical status can lead to degeneration, also brings risk into estimation of importance sampling, so, the drawback regarding this method is degeneration. In fact, frequent importance sampling has in it more degeneration compared to normal importance sampling because the resampling phase is repeated several times in frequent importance sampling. The percent of particles to be formed between two iterations from resampling algorithm can be low.it is obvious that the probability of such case might be increase by the increase of iterations. The implication of such population degeneration is that the number of main sample branches decreases rapidly. In such cases, if the proposal distributions are based on the generated values, then it might be possible in final output that there is symmetric status or at least it might increase the variance of estimators. Also, similar to normal importance ( sampling there is a risk that ) weights lead to the infinite variances (Celeux et al., 2006). There are other similarities between PMC and proposal distributions in particle systems in Gilks and Berzuini (2006) since these authors have taken into account the frequent samples through resampling phase based on the importance weights. However, the big difference (despite their dynamic statues difference for mobile target distributions) is that they remain in the domain of MCMC because they do the resampling phase prior to the proposal distributions running. So, these authors had to use the Markov condition changing core for the given stationary distributions. Also, there is a similarity with Chopin (2002) which used the frequent importance sampling by the proposal distributions. This case is a special one in PMC on the Bayesian framework so that the proposaldistributions of are the corresponding posterior distributions accompanied by element from the observed class. As mentioned above, one of the prominent attributes of PMC method is choosing proposal distribution of freely because of the MCMC framework ignoring. In fact, a Metropolis-Hastings for each point from the posterior sample generates one parallel MCMC sampler which is simply convergent to the target distribution of within the distribution without reforming the importance resampling. Similarly, a Metropolis-Hastings accept ( phase for the whole vector of ) is convergent to. The advantage of generating an asymptote estimation over i.i.d sample is declined by the probability that the accepting is decreased as power approximately. ( Hence, in each iteration points are chosen according to their ) importance weight in PMC. Sometimes, it occurs that the Metropolis-Hastingsmethod doesnot work properly based on this proposal distribution while a PMC algorithm generates correct and valid responses (Cappé et al., 2004). So, PMC framework provides simpler structure of adaptive methods compared to MCMC in fact, as long as the importance sampling strategies are considered in prior section of MCMC, the MCMC context is less appropriate for adaptive algorithms since adaptability invalidates the essence of continues Markov. Because of this, it needs more studies on convergence to make the argotic attribute. For more information on this, see example Anderieu and Robert (2001). In PMC methods, the argotic attribute is not considered as a solution since the validity of this algorithm has been obtained by means of describing the importance sampling. The generated samples by PMC can exploit the importance sampling outputs in each iteration and in this way it does not require stop rules the same as MCMC samples (Robert and Casella, 2004). Anyway, as it was described with constant estimation in Robert and Casella(2004), one would exploit from all of the continuum of samples both for proposal distributions and for estimation.in order to use the samples continuum, one does not need constant storing pace for all of the generated samples because estimations like that of (3) can be updated dynamically. In addition, the possibility of stimulations exploitation implies that the sample value is not necessarily large because the effective simulation volume is. The last point is that the number of points within the sample in iterations is not necessarily constant. As (Chopin, 2002), one would increase the number of points within the sample when the algorithm is fixed in one status. In order to survey this method, we analyze a simple instance whose answer is known. Example 1.2 suppose the target of solving integral is as follows: = 525

5 Where h( ) = and ( ) = 1. Also, assumethat one random sample is available from ( )distribution. The estimation of this integral using Population Monte Carlo would be as follows: = = I = ( ) ( ) ( ) ( ) The resulting estimation will be as figure 1 for = 100 iteration. We know = As it is seen, the integral value is convergent to which is in consistency with main integral and with the obtained result from Monte Carlo method. Figure 1. PMC Estimator with 1000 iterations Example 1.3 suppose X~τ(ν, θ, σ ) is the density distribution: π(x) = Γ σ νπγ(ν 2) the aim is calculating the following integral: I = [h(x)] = (1 + (x θ) νσ ) ( ) x 1 + (x 3) π(x)dx Suppose θ = 0, σ = 1and ν = 12without reducing the generality of problem. We use Population Monte Carlo method to solve the above integral. In doing so, we assume Exp(1) function as the importance function (Robert and Casella, 2004). The integral formed by such proposal distribution would be as follows: x e I = π(x)e dx 1 + (x 3) According to the importance sampling method, the estimator I would be as follows: I = ϱ( ) h(x ( ) ) ϱ ( ) Where (.) is t-student distribution, (.) is the exponential distribution including parameter 1 and h = 51+( 3)2. As is obvious, the value of estimator decreases and converges to a constant value by increasingof the number of iterations. The resulting estimation for =2000 would be as figure

6 Figure 2. PMC Estimator with 2000 iterations REFERENCES Anderieu C, Robert CP Controlled Markov Chain Monte Carlo Methods for Optimal Sampling. Université Paris Dauphine, Cappé O, Guillin A, Marin JM, Robert C Population Monte Carlo. Journal of Computational and GraphicalStatististics, 13(4): Celeux G, Marin JM, Robert CP Iterated Importance Sampling in Missing Data Problems. Computational Statistics and Data Analysis, 50(12) Chopin N A Sequential Particle Filter Method for Static Models. Biometrica, 89: , Christian PR, George C Monte Carlo Statistical Methods, Springer, Second Edition, Douc R, Guillin A, Marin JM, Robert CP Minimum Variance Importance Sampling via Population Monte Carlo. Technical report, University Paris Dauphine. Gilks W, Berzuini C Following a moving target-monte Carlo inference for dynamic Bayesian models. J. Royal Statist. Soc. Series B, 63(1): , Mengersen K, Robert C Iid sampling with self-avoiding particle filters: the pinball sampler. In Bernardo, J., Bayarri, M., Berger, J., Dawid, A., Heckerman, D., Smith, A., and West, M., editors. Bayesian Statistics, 7. Oxford University Press, Oxford. Warnes, G The Normal kernel coupler: An adaptive Markov Chain Monte Carlo Method for efficiently sampling from multi-modal distributions. Technical Report 395, Univ. if Washington, 527