On empirical cumulant generating functions of code lengths for individual sequences

We consider the problem of lossless compression of individual sequences using finite-state (FS) machines, from the perspective of the best achievable empirical cumulant generating function (CGF) of the code length, i.e., the normalized logarithm of the empirical average of the exponentiated code length. Since the probabilistic CGF is minimized in terms of the R\'enyi entropy of the source, one of the motivations of this study is to derive an individual-sequence analogue of the R\'enyi entropy, in the same way that the FS compressibility is the individual-sequence counterpart of the Shannon entropy. We consider the CGF of the code-length both from the perspective of fixed-to-variable (F-V) length coding and the perspective of variable-to-variable (V-V) length coding, where the latter turns out to yield a better result, that coincides with the FS compressibility. We also extend our results to compression with side information, available at both the encoder and decoder. In this case, the V-V version no longer coincides with the FS compressibility, but results in a different complexity measure.Comment: 15 pages; submitted for publicatio

Optimum estimation via gradients of partition functions and information measures: a statistical-mechanical perspective

In continuation to a recent work on the statistical--mechanical analysis of minimum mean square error (MMSE) estimation in Gaussian noise via its relation to the mutual information (the I-MMSE relation), here we propose a simple and more direct relationship between optimum estimation and certain information measures (e.g., the information density and the Fisher information), which can be viewed as partition functions and hence are amenable to analysis using statistical--mechanical techniques. The proposed approach has several advantages, most notably, its applicability to general sources and channels, as opposed to the I-MMSE relation and its variants which hold only for certain classes of channels (e.g., additive white Gaussian noise channels). We then demonstrate the derivation of the conditional mean estimator and the MMSE in a few examples. Two of these examples turn out to be generalizable to a fairly wide class of sources and channels. For this class, the proposed approach is shown to yield an approximate conditional mean estimator and an MMSE formula that has the flavor of a single-letter expression. We also show how our approach can easily be generalized to situations of mismatched estimation.Comment: 21 pages; submitted to the IEEE Transactions on Information Theor