Transport-entropy inequalities and curvature in discrete-space Markov chains
نویسندگان
چکیده
Let G = (Ω, E) be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Zt} on Ω whose kernel p satisfies p(x, y) > 0 ⇒ {x, y} ∈ E for every x, y ∈ Ω. In words, transitions only occur between neighboring points of the graph. Suppose further that (Ω, p, d) has coarse Ricci curvature at least 1/α in the sense of Ollivier: For all x, y ∈ Ω, it holds that W1(Z1 | {Z0 = x}, Z1 | {Z0 = y}) ≤ (
منابع مشابه
Curvature and transport inequalities for Markov chains in discrete spaces
We study various transport-information inequalities under three di erent notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery [4], the exponential curvature-dimension condition of Bauer et al. [6] and the coarse Ricci curvature of Ollivier [38]. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an L1 tran...
متن کاملRelative Entropy Rate between a Markov Chain and Its Corresponding Hidden Markov Chain
In this paper we study the relative entropy rate between a homogeneous Markov chain and a hidden Markov chain defined by observing the output of a discrete stochastic channel whose input is the finite state space homogeneous stationary Markov chain. For this purpose, we obtain the relative entropy between two finite subsequences of above mentioned chains with the help of the definition of...
متن کاملThe Rate of Rényi Entropy for Irreducible Markov Chains
In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.
متن کاملENTROPY FOR DTMC SIS EPIDEMIC MODEL
In this paper at rst, a history of mathematical models is given.Next, some basic information about random variables, stochastic processesand Markov chains is introduced. As follows, the entropy for a discrete timeMarkov process is mentioned. After that, the entropy for SIS stochastic modelsis computed, and it is proved that an epidemic will be disappeared after a longtime.
متن کاملTransportation-information Inequalities for Markov Processes (ii) : Relations with Other Functional Inequalities
We continue our investigation on the transportation-information inequalities WpI for a symmetric markov process, introduced and studied in [14]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure μ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion pr...
متن کامل