A Bayesian Bootstrap for Finite State Markov Chains 1017

Bernoulli, 2006

Les documents de travail ne reflètent pas la position de l'INSEE et n'engagent que leurs auteurs.

Computational Statistics & Data Analysis, 2008

2003

In this paper, we show how the original Bootstrap method introduced by Datta & McCormick (1993), namely the regenerationbased Bootstrap, for approximating the sampling distribution of sample mean statistics in the atomic Markovian setting may be modified, so as to be second order correct. We prove that the drawback of the original construction mainly relies on a wrong estimation of the skewness of the sampling distribution and that it is possible to correct it by suitable standardization of the regeneration-based bootstrap statistic and recentering of the bootstrap distribution. An asymptotic result establishing the second order accuracy of this bootstrap estimate up to O(n −1 log(n)) (close to the rate obtained in an i.i.d. setting) is also stated under weak moment assumptions.

Cornell University - arXiv, 2019

Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to the tasks of (i) determining MCMC burn-in, (ii) comparing different MCMC algorithms with the same target, and (iii) comparing exact and approximate MCMC. Lastly, we (iv) assess the bias of sequential Monte Carlo and self-normalized importance samplers.

Annals of Statistics, 1996

The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a self-contained proof of the convergence of this method in general state spaces under conditions that are easy to verify.

Econometrica, 2003

This study compares the restricted and unrestricted methods of bootstrap data generating processes (DGPs) on statistical inference. It used hypothetical datasets simulated from normal distribution with different ability levels. Data were analyzed using different bootstrap DGPs. In practice, it is advisable to use the restricted parametric bootstrap DGP models and thereafter, check the kernel density of the empirical distributions that are close to normal (at least not too skewed). In fact, 21600 scenarios were replicated 200 times using bootstrap DGPs and kernel density methods. This analysis was carried out using R-statistical package. The results show that in a situation where the distribution of a test is skewed, all the scores need to be taken into account, no matter how small the sample size and the bootstrap level are. Across all the conditions considered, models HR5UR and HPN5UR yielded much larger bias and standard error while the smallest bias values were associated with models HR5R (0.0619) and HPN5R (0.0624). The result confirms the fact that bootstrap DGPs are very vital in statistical inference.

P m (x, ·) − Π(·) T V → 0 as m → ∞.

2020

Note this is a random variable with expected value π(f) (i.e. the estimator is unbiased) and standard deviation of order O(1/ √ N). Then by CLT, the errorπ(f) − π(f) will have a limiting normal distribution as N → ∞. Therefore we can compute π(f) by computing samples (plus some regression techniques?). But the problem is if π u is complicated, then it is very difficult to simulate i.i.d. random variables from π(•). The MCMC solution is to construct a Markov chain on X which has π(•) as a stationary distribution, i.e. X π(dx)P (x, dy) = π(dy) Then for large n the distribution of X n will be approximately stationary. We can set Z 1 = X n and get Z 2 , Z 3 ,. .. , Z n repeatedly. Remark. In practice instead of starting a fresh Markov chain every time we take the successive X n 's, for example, (N − B) −1 N i=B+1 f (X i). We tend to ignore the dependence problem as many of the mathematical issues are similar in either implementation. Remark. We have other ways of estimation, such as "rejection sampling" and "importance sampling". But MCMC algorithms is applied most widely. 2 MCMC and its construction This section will explain how MCMC algorithm is constructed. Now we introduce reversibility. Definition. A Markov Chain on state space X is reversible with respect to a probability distribution π(•) on X , if π(dx)P (x, dy) = π(dy)P (y, dx), x, y ∈ X Proposition. A Markov Chain is reversible with respect to π(•), then π(•) is the stationary distribution for the chain. Proof. By reversibility, we have x∈X π(dx)P (x, dy) = x∈X π(dy)P (y, dx) = π(dy) x∈X P (x, dy) = π(dy) Now the simplest way to construct a MCMC algorithm which satisfies reversibility is using Metropolis-Hastings algorithm. 2.1 The Metropolis-Hastings Algorithm. Suppose that π(•) has a (possibly unnormalized) density π u. Let Q(x, •) be essentially any other Markov Chain, whose transitions also have a (possibly unnormalized) density, i.e. Q(x, dy) ∝ q(x, y)dy. First choose some X 0. Then given X n , generate a proposal Y n+1 from Q(X n , •). In the meantime we flip a independent bias coin with probability of heads equals to α(X n , Y n+1), where α(x, y) = min 1, π u (y)q(y, x) π u (x)q(x, y) , π(x)q(x, y) = 0 And α(x, y) = 1 when π(x)q(x, y) = 0. Then if the coin is heads, we accept the proposal and set X n+1 = Y n+1. If the coin is tails, then we reject the proposal and set X n+1 = X n. Then we replace n by n + 1 and repeat. The reason we take α(x, y) as above is explain as follow. Proposition. The Metropolis-Hastings Algorithm produces a Markov Chain {X n } which is reversible with respect to π(•). Proof. We want to show for any x, y ∈ X , π(dx)P (x, dy) = π(dy)P (y, dx) whereȲ i = 1 J j Y ij. The Gibbs sampler then proceeds by updating the K + 3 variables according to the above conditional distributions. This is feasible since the conditional distributions are all easily simulated (IG and N).

2010

Simulation is an interesting alternative to solve Markovian models. However, when compared to analytical and numerical solutions it suffers from a lack of precision in the results due to the very nature of simulation, which is the choice of samples through pseudorandom generation. This paper proposes a different way to simulate Markovian models by using a Bootstrap-based statistical method to minimize the effect of sample choices. The effectiveness of the proposed method, called Bootstrap simulation, is compared to the numerical solution results for a set of examples described using Stochastic Automata Networks modeling formalism.

Bootstrapping time series is one of the most acknowledged tools to make forecasts and study the statistical properties of an evolutive phenomenon. The idea underlying this procedure is to replicate the phenomenon on the basis of an observed sample. One of the most important classes of bootstrap procedures is based on the assumption that the sampled phenomenon evolves according to a Markov chain. Such an assumption does not apply when the process takes values in a continuous set, as frequently happens for time series related to economic and financial variables. In this paper we apply Markov chain theory for bootstrapping continuous processes, relying on the idea of discretizing the support of the process and suggesting Markov chains of order k to model the evolution of the time series under study. The difficulty of this approach is that, even for small k, the number of rows of the transition probability matrix is too large, and this leads to a bootstrap procedure of high complexity. In many practical cases such complexity is not fully justified by the information really required to replicate a phenomenon satisfactorily. In this paper we propose a methodology to reduce the number of rows without loosing "too much" information on the process evolution. This requires a clustering of the rows that preserves as much as possible the "law" that originally generated the process. The novel aspect of our work is the use of Mixed Integer Linear Programming for formulating and solving the problem of clustering similar rows in the original transition probability matrix. Even if it is well known that this problem is computationally hard, in our application medium size real-life instances were solved efficiently. Our empirical analysis, which is done on two time series of prices from the German and the Spanish electricity markets, shows that the use of the aggregated transition probability matrix does not affect the bootstrapping procedure, since the characteristic features of the original series are maintained in the resampled ones.