scholarly journals Extended Laplace principle for empirical measures of a Markov chain

2019 ◽  
Vol 51 (01) ◽  
pp. 136-167 ◽  
Author(s):  
Stephan Eckstein
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Large Deviation ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Large Deviations Principle ◽  
Empirical Measures ◽  
Leibler Divergence ◽  
Laplace Principle

AbstractWe consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp (f). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance. The setting and proof of the extended Laplace principle are based on the weak convergence approach to large deviations by Dupuis and Ellis (2011).

scholarly journals A non-exponential extension of Sanov’s theorem via convex duality

2020 ◽  
Vol 52 (1) ◽  
pp. 61-101
Author(s):  
Daniel Lacker
Keyword(s):  
Optimal Transport ◽  
Large Deviation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Empirical Measures ◽  
Empirical Distributions

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains

1990 ◽  
Vol 27 (1) ◽  
pp. 44-59 ◽  
Author(s):  
James A Bucklew ◽  
Peter Ney ◽  
John S. Sadowsky
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Large Deviation ◽  
Homogeneous Markov Chain ◽  
Underlying Distribution ◽  
Monte Carlo Simulation Technique ◽  
Large Deviations Theory

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.

scholarly journals Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 573 ◽  
Author(s):  
Rodrigo Cofré ◽  
Cesar Maldonado ◽  
Fernando Rosas
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Maximum Entropy ◽  
Statistical Physics ◽  
Large Deviation ◽  
Thermodynamic Formalism ◽  
Spike Trains ◽  
Statistical Fluctuation ◽  
Inference Problem

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. To find the maximum entropy Markov chain, we use the thermodynamic formalism, which provides insightful connections with statistical physics and thermodynamics from which large deviations properties arise naturally. We provide an accessible introduction to the maximum entropy Markov chain inference problem and large deviations theory to the community of computational neuroscience, avoiding some technicalities while preserving the core ideas and intuitions. We review large deviations techniques useful in spike train statistics to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability, and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains

1990 ◽  
Vol 27 (01) ◽  
pp. 44-59 ◽  
Author(s):  
James A Bucklew ◽  
Peter Ney ◽  
John S. Sadowsky
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Large Deviation ◽  
Homogeneous Markov Chain ◽  
Underlying Distribution ◽  
Monte Carlo Simulation Technique ◽  
Large Deviations Theory

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.

A large deviations heuristic made precise

2000 ◽  
Vol 128 (3) ◽  
pp. 561-569 ◽  
Author(s):  
NEIL O'CONNELL
Keyword(s):  
Large Deviations ◽  
Bin Packing ◽  
Symmetric Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Packing Problem ◽  
Rate Function ◽  
Empirical Measures ◽  
Underlying Distribution ◽  
Uniformly Lipschitz

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.

Stochastic scrabble: large deviations for sequences with scores

1988 ◽  
Vol 25 (1) ◽  
pp. 106-119 ◽  
Author(s):  
Richard Arratia ◽  
Pricilla Morris ◽  
Michael S. Waterman
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Additive Functional ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Similar Treatment ◽  
Large Numbers ◽  
Large Deviation Result

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

scholarly journals Large deviations for stochastic integrodifferential equations of the Itô type with multiple randomness

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 473-487 ◽  
Author(s):  
A. Haseena ◽  
M. Suvinthra ◽  
N. Annapoorani
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Finite Number ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Integrodifferential Equations ◽  
Multiplicative Noises ◽  
Deviation Principle ◽  
Laplace Principle ◽  
Weak Convergence Approach

A Freidlin-Wentzell type large deviation principle is derived for a class of It? type stochastic integrodifferential equations driven by a finite number of multiplicative noises of the Gaussian type. The weak convergence approach is used here to prove the Laplace principle, equivalently large deviation principle.

Large deviations for longest runs in Markov chains

2020 ◽  
Vol 26 (2) ◽  
pp. 309-314
Author(s):  
Zhenxia Liu ◽  
Yurong Zhu
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Markov Chains ◽  
Present Note ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Large Deviation Principles ◽  
Success Run ◽  
Longest Success Run ◽  
Math Statist

AbstractWe continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

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