There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

On robust estimation of the location parameter

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

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Prediction, learning, and games

An overview of statistical and information-theoretic aspects of hidden Markov processes (HMPs) is presented. An HMP is a discrete-time finite-state homogeneous Markov chain observed through a discrete-time memoryless invariant channel. In recent years, the work of Baum and Petrie (1966) on finite-state finite-alphabet HMPs was expanded to HMPs with finite as well as continuous state spaces and a general alphabet. In particular, statistical properties and ergodic theorems for relative entropy densities of HMPs were developed. Consistency and asymptotic normality of the maximum-likelihood (ML) parameter estimator were proved under some mild conditions. Similar results were established for switching autoregressive processes. These processes generalize HMPs. New algorithms were developed for estimating the state, parameter, and order of an HMP, for universal coding and classification of HMPs, and for universal decoding of hidden Markov channels. These and other related topics are reviewed.

http://www.isib.cnr.it/control/finesso/Ephraim_hmm.pdf

Hidden Markov processes

This paper consists of an overview on universal prediction from an information-theoretic perspective. Special attention is given to the notion of probability assignment under the self-information loss function, which is directly related to the theory of universal data compression. Both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings.

Universal prediction

Reliable transmission over a discrete-time memoryless channel with a decoding metric that is not necessarily matched to the channel (mismatched decoding) is considered. It is assumed that the encoder knows both the true channel and the decoding metric. The lower bound on the highest achievable rate found by Csiszar and Korner (1981) and by Hui (1983) for DMC's, hereafter denoted C/sub LM/, is shown to bear some interesting information-theoretic meanings. The bound C/sub LM/ turns out to be the highest achievable rate in the random coding sense, namely, the random coding capacity for mismatched decoding. It is also demonstrated that the /spl epsiv/-capacity associated with mismatched decoding cannot exceed C/sub LM/. New bounds and some properties of C/sub LM/ are established and used to find relations to the generalized mutual information and to the generalized cutoff rate. The expression for C/sub LM/ is extended to a certain class of memoryless channels with continuous input and output alphabets, and is used to calculate C/sub LM/ explicitly for several examples of theoretical and practical interest. Finally, it is demonstrated that in contrast to the classical matched decoding case, here, under the mismatched decoding regime, the highest achievable rate depends on whether the performance criterion is the bit error rate or the message error probability and whether the coding strategy is deterministic or randomized. >

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On information rates for mismatched decoders

The problem of predicting the next outcome of an individual binary sequence using finite memory is considered. The finite-state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite-state (FS) predictor. It is proven that this FS predictability can be achieved by universal sequential prediction schemes. An efficient prediction procedure based on the incremental parsing procedure of the Lempel-Ziv data compression algorithm is shown to achieve asymptotically the FS predictability. Some relations between compressibility and predictability are discussed, and the predictability is proposed as an additional measure of the complexity of a sequence. >

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Universal prediction of individual sequences

The relation between the entropy of a discrete random variable and the minimum attainable probability of error made in guessing its value is examined. While Fano's inequality provides a tight lower bound on the error probability in terms of the entropy, the present authors derive a converse result/spl mdash/a tight upper bound on the minimal error probability in terms of the entropy. Both bounds are sharp, and can draw a relation, as well, between the error probability for the maximum a posteriori (MAP) rule, and the conditional entropy (equivocation), which is a useful uncertainty measure in several applications. Combining this relation and the classical channel coding theorem, the authors present a channel coding theorem for the equivocation which, unlike the channel coding theorem for error probability, is meaningful at all rates. This theorem is proved directly for DMCs, and from this proof it is further concluded that for R/spl ges/C the equivocation achieves its minimal value of R/spl minus/C at the rate of n/sup 1/spl sol/2/ where n is the block length. >

Neri Merhav

Papers

Hidden Markov processes

Universal prediction

On information rates for mismatched decoders

Universal prediction of individual sequences

Relations between entropy and error probability