arXiv subjects

Krzysztof Latuszynski

Publications and source records attributed to Krzysztof Latuszynski.

14 recordsLinked to original sources

arXiv2025

Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo

We consider adaptive increasingly rare Markov chain Monte Carlo (MCMC) algorithms, which are adaptive MCMC methods, where the adaptation concerning the "past'' happens less and less frequently over time. Under a contraction assumption with respect to a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is "almost'' the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed.

math.NA↗

arXiv2020

From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains

Given a $p$-coin that lands heads with unknown probability $p$, we wish to produce an $f(p)$-coin for a given function $f: (0,1) \rightarrow (0,1)$. This problem is commonly known as the Bernoulli Factory and results on its solvability and complexity have already been obtained. Nevertheless, generic ways to design a practical Bernoulli Factory for a given function $f$ exist only in a few special cases. We present a constructive way to build an efficient Bernoulli Factory when $f(p)$ is a rational function with coefficients in $\mathbb{R}$. Moreover, we extend the Bernoulli Factory problem to a more general setting where we have access to an $m$-sided die and we wish to roll a $v$-sided one; i.e., we consider rational functions between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains - a task that we show can be achieved using perfect simulation techniques with the original $m$-sided die as the only source of randomness. In the Bernoulli Factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in $p$. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.

math.PR↗

arXiv2018

Adapting The Gibbs Sampler

The popularity of Adaptive MCMC has been fueled on the one hand by its success in applications, and on the other hand, by mathematically appealing and computationally straightforward optimisation criteria for the Metropolis algorithm acceptance rate (and, equivalently, proposal scale). Similarly principled and operational criteria for optimising the selection probabilities of the Random Scan Gibbs Sampler have not been devised to date. In the present work, we close this gap and develop a general purpose Adaptive Random Scan Gibbs Sampler that adapts the selection probabilities. The adaptation is guided by optimising the $L_2-$spectral gap for the target's Gaussian analogue, gradually, as target's global covariance is learned by the sampler. The additional computational cost of the adaptation represents a small fraction of the total simulation effort. ` We present a number of moderately- and high-dimensional examples, including truncated Gaussians, Bayesian Hierarchical Models and Hidden Markov Models, where significant computational gains are empirically observed for both, Adaptive Gibbs, and Adaptive Metropolis within Adaptive Gibbs version of the algorithm. We argue that Adaptive Random Scan Gibbs Samplers can be routinely implemented and substantial computational gains will be observed across many typical Gibbs sampling problems. We shall give conditions under which ergodicity of the adaptive algorithms can be established.

stat.CO↗

arXiv2018

Air Markov Chain Monte Carlo

We introduce a class of Adapted Increasingly Rarely Markov Chain Monte Carlo (AirMCMC) algorithms where the underlying Markov kernel is allowed to be changed based on the whole available chain output but only at specific time points separated by an increasing number of iterations. The main motivation is the ease of analysis of such algorithms. Under the assumption of either simultaneous or (weaker) local simultaneous geometric drift condition, or simultaneous polynomial drift we prove the $L_2-$convergence, Weak and Strong Laws of Large Numbers (WLLN, SLLN), Central Limit Theorem (CLT), and discuss how our approach extends the existing results. We argue that many of the known Adaptive MCMC algorithms may be transformed into the corresponding Air versions, and provide an empirical evidence that performance of the Air version stays virtually the same.

stat.CO↗

arXiv2014

Individual adaptation: an adaptive MCMC scheme for variable selection problems

The increasing size of data sets has lead to variable selection in regression becoming increasingly important. Bayesian approaches are attractive since they allow uncertainty about the choice of variables to be formally included in the analysis. The application of fully Bayesian variable selection methods to large data sets is computationally challenging. We describe an adaptive Markov chain Monte Carlo approach called Individual Adaptation which adjusts a general proposal to the data. We show that the algorithm is ergodic and discuss its use within parallel tempering and sequential Monte Carlo approaches. We illustrate the use of the method on two data sets including a gene expression analysis with 22 577 variables.

stat.CO↗

arXiv2014

Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We then prove that a recently introduced Markov kernel in this setting can inherit variance bounding and geometric ergodicity from its intractable Metropolis--Hastings counterpart, under reasonably weak and manageable conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results on an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, time-homogeneous, pure jump Markov process. We also supply two general theorems, one of which provides a simple sufficient condition for lack of variance bounding for reversible kernels and the other provides a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.

stat.ME↗

arXiv2013

The Containment Condition and AdapFail algorithms

This short note investigates convergence of adaptive MCMC algorithms, i.e.\ algorithms which modify the Markov chain update probabilities on the fly. We focus on the Containment condition introduced in \cite{roberts2007coupling}. We show that if the Containment condition is \emph{not} satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then \emph{any} nonadaptive ergodic MCMC algorithm. We call such algorithms \texttt{AdapFail}, and conclude that they should not be used.

stat.CO↗

arXiv2011

CLTs and asymptotic variance of time-sampled Markov chains

For a Markov transition kernel $P$ and a probability distribution $ μ$ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel $P_μ = \sum_k μ(k)P^k.$ In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance.

math.PR↗

arXiv2011

Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques

The Nummellin's split chain construction allows to decompose a Markov chain Monte Carlo (MCMC) trajectory into i.i.d. "excursions". RegenerativeMCMC algorithms based on this technique use a random number of samples. They have been proposed as a promising alternative to usual fixed length simulation [25, 33, 14]. In this note we derive nonasymptotic bounds on the mean square error (MSE) of regenerative MCMC estimates via techniques of renewal theory and sequential statistics. These results are applied to costruct confidence intervals. We then focus on two cases of particular interest: chains satisfying the Doeblin condition and a geometric drift condition. Available explicit nonasymptotic results are compared for different schemes of MCMC simulation.

stat.CO↗

arXiv2010

Adaptive Gibbs samplers

We consider various versions of adaptive Gibbs and Metropolis within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions.

stat.CO↗

arXiv2009

Simulating Events of Unknown Probabilities via Reverse Time Martingales

Assume that one aims to simulate an event of unknown probability $s\in (0,1)$ which is uniquely determined, however only its approximations can be obtained using a finite computational effort. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non-linear diffusions, second, the celebrated Bernoulli factory problem of generating an $f(p)-$coin given a sequence $X_1,X_2,...$ of independent tosses of a $p-$coin (with known $f$ and unknown $p$). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability $s.$ In the case of diffusions, we obtain the algorithm of \cite{BeskosRobertsEA1} as a specific instance of the generic framework developed here. In the case of the Bernoulli factory, our work offers a statistical understanding of the Nacu-Peres algorithm for $f(p) = \min\{2p, 1-2\varepsilon\}$ (which is central to the general question) and allows for its immediate implementation that avoids algorithmic difficulties of the original version.

stat.CO↗

arXiv2009

Rigorous confidence bounds for MCMC under a geometric drift condition

We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function $f:\stany \to R,$ let $I=\int_{\stany} f(x) π(x) dx$ be the value of interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq 1-α.$$ The bounds depend only and explicitly on drift parameters, on the $V-$norm of $f,$ where $V$ is the drift function and on precision and confidence parameters $\varepsilon, α.$ Next we analyse an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example.

stat.ME↗

arXiv2009

Nonasymptotic bounds on the estimation error for regenerative MCMC algorithms

MCMC methods are used in Bayesian statistics not only to sample from posterior distributions but also to estimate expectations. Underlying functions are most often defined on a continuous state space and can be unbounded. We consider a regenerative setting and Monte Carlo estimators based on i.i.d. blocks of a Markov chain trajectory. The main result is an inequality for the mean square error. We also consider confidence bounds. We first derive the results in terms of the asymptotic variance and then bound the asymptotic variance for both uniformly ergodic and geometrically ergodic Markov chains.

stat.ME↗

arXiv2009

Regeneration and Fixed-Width Analysis of Markov Chain Monte Carlo Algorithms

In the thesis we take the split chain approach to analyzing Markov chains and use it to establish fixed-width results for estimators obtained via Markov chain Monte Carlo procedures (MCMC). Theoretical results include necessary and sufficient conditions in terms of regeneration for central limit theorems for ergodic Markov chains and a regenerative proof of a CLT version for uniformly ergodic Markov chains with $E_πf^2< \infty.$ To obtain asymptotic confidence intervals for MCMC estimators, strongly consistent estimators of the asymptotic variance are essential. We relax assumptions required to obtain such estimators. Moreover, under a drift condition, nonasymptotic fixed-width results for MCMC estimators for a general state space setting (not necessarily compact) and not necessarily bounded target function $f$ are obtained. The last chapter is devoted to the idea of adaptive Monte Carlo simulation and provides convergence results and law of large numbers for adaptive procedures under path-stability condition for transition kernels.

stat.ME↗