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

Analysis of Mismatched Estimation Errors Using Gradients of Partition Functions

We consider the problem of signal estimation (denoising) from a statistical-mechanical perspective, in continuation to a recent work on the analysis of mean-square error (MSE) estimation using a direct relationship between optimum estimation and certain partition functions. The paper consists of essentially two parts. In the first part, using the aforementioned relationship, we derive single-letter expressions of the mismatched MSE of a codeword (from a randomly selected code), corrupted by a Gaussian vector channel. In the second part, we provide several examples to demonstrate phase transitions in the behavior of the MSE. These examples enable us to understand more deeply and to gather intuition regarding the roles of the real and the mismatched probability measures in creating these phase transitions.Comment: 58 pages;Submitted to IEEE Trans. on Information Theor

Achievable Outage Rates with Improved Decoding of Bicm Multiband Ofdm Under Channel Estimation Errors

We consider the decoding of bit interleaved coded modulation (BICM) applied to multiband OFDM for practical scenarios where only a noisy (possibly very bad) estimate of the channel is available at the receiver. First, a decoding metric based on the channel it a posteriori probability density, conditioned on the channel estimate is derived and used for decoding BICM multiband OFDM. Then, we characterize the limits of reliable information rates in terms of the maximal achievable outage rates associated to the proposed metric. We also compare our results with the outage rates of a system using a theoretical decoder. Our results are useful for designing a communication system where a prescribed quality of service (QoS), in terms of achievable target rates with small error probability, must be satisfied even in the presence of imperfect channel estimation. Numerical results over both realistic UWB and theoretical Rayleigh fading channels show that the proposed method provides significant gain in terms of BER and outage rates compared to the classical mismatched detector, without introducing any additional complexity

Pointwise Relations between Information and Estimation in Gaussian Noise

Many of the classical and recent relations between information and estimation in the presence of Gaussian noise can be viewed as identities between expectations of random quantities. These include the I-MMSE relationship of Guo et al.; the relative entropy and mismatched estimation relationship of Verd\'{u}; the relationship between causal estimation and mutual information of Duncan, and its extension to the presence of feedback by Kadota et al.; the relationship between causal and non-casual estimation of Guo et al., and its mismatched version of Weissman. We dispense with the expectations and explore the nature of the pointwise relations between the respective random quantities. The pointwise relations that we find are as succinctly stated as - and give considerable insight into - the original expectation identities. As an illustration of our results, consider Duncan's 1970 discovery that the mutual information is equal to the causal MMSE in the AWGN channel, which can equivalently be expressed saying that the difference between the input-output information density and half the causal estimation error is a zero mean random variable (regardless of the distribution of the channel input). We characterize this random variable explicitly, rather than merely its expectation. Classical estimation and information theoretic quantities emerge with new and surprising roles. For example, the variance of this random variable turns out to be given by the causal MMSE (which, in turn, is equal to the mutual information by Duncan's result).Comment: 31 pages, 2 figures, submitted to IEEE Transactions on Information Theor