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Register a new userA Likelihood Ratio Approach for Probabilistic Inequalities
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Xinjia Chen
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Authors
Xinjia Chen
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We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities such as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment generating functions, tightest possible concentration inequalities may be readily derived by the proposed approach. We have derived new concentration inequalities using the proposed approach, which cannot be obtained by the classical approach based on moment generating functions.
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We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffdings inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions to obtain multivariate concentration inequalities.
We explore the applications of our previously established likelihood-ratio method for deriving concentration inequalities for a wide variety of univariate and multivariate distributions. New concentration inequalities for various distributions are developed without the idea of minimizing moment generating functions.
We consider a discrete-time Markov chain $boldsymbol{Phi}$ on a general state-space ${sf X}$, whose transition probabilities are parameterized by a real-valued vector $boldsymbol{theta}$. Under the assumption that $boldsymbol{Phi}$ is geometrically ergodic with corresponding stationary distribution $pi(boldsymbol{theta})$, we are interested in estimating the gradient $ abla alpha(boldsymbol{theta})$ of the steady-state expectation $$alpha(boldsymbol{theta}) = pi( boldsymbol{theta}) f.$$ To this end, we first give sufficient conditions for the differentiability of $alpha(boldsymbol{theta})$ and for the calculation of its gradient via a sequence of finite horizon expectations. We then propose two different likelihood ratio estimators and analyze their limiting behavior.
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheegers inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheegers inequality, which is a $L_{p}$-Poincar{e} inequality for $p=2$, to any real $pgeq 1$.
This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see cite{Lasry_Lions}, and by Huang, Caines, and Malham{e}, see cite{Huang}. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.
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