A Statistical Mechanics Approach to Large Deviation Theorems

Chernoo bounds and related large deviation bounds have a wide variety of applications in statistics and learning theory. This paper proves that for any real-valued random variable X the probability of a deviation to value x is bounded by e S(x) where S(x) is the entropy at energy x of a physical system corresponding to the variable X. It is a well known fact of statistical mechanics that entropy is equal to a double integral of the reciprocal of energy variance. So we get a general bound on large deviation probabilities in terms of variance over a range of temperatures. This greatly simpliies the derivation of Chernoo and Hoeeding bounds and leads immediately to a variety of apparently new large deviation theorems.