arXiv subjects

Edward D. Weinberger

Publications and source records attributed to Edward D. Weinberger.

5 recordsLinked to original sources

arXiv2026

Pragmatic Information, Computation, and the Efficient Market Hypothesis

The efficient market hypothesis, that prices reflect all available information, imputes a meaning to market moving information, a view of information foreign to standard information theory. Here, after reviewing properties that make a proposed formula for ``pragmatic information" a plausible measure of meaning, we consider the role that the receiver's position in the machine hierarchy corresponding to the Chomsky hierarchy play in extracting this meaning. We show that a receiver at a given level in the hierarchy may be unable to extract pragmatic information from a message because it appears random, yet a receiver at a higher level in the hierarchy finds the message perfectly intelligible. Also, the maximum processing rates for messages of different levels of hierarchy serve as a kind of channel capacity, leading to a tradeoff between the amount of pragmatic information extracted and the extraction time. All of the above suggests a recasting of market efficiency in terms of ``computational efficiency'', i.e. the question of whether a market appears efficient to a participant with a given computational endowment. Successful trading strategies implementable as finite state machines, the lowest level in the hierarchy, imply departures from ``finite state efficiency''. We show, via a stylized example, how pragmatic information can characterize the ensuing approach to finite state efficiency. We also show that actual market dynamics can be more computationally intractable than any finite state machine can process by proving the PSPACE-completeness of the processing of the smart order routing systems of major brokerage firms. We conclude that computational efficiency is the norm and pragmatic information can characterize any departure from it.

cs.CE↗

arXiv2026

Towards a Theory of Pragmatic Information

Standard information theory says nothing about how much meaning is conveyed by a message. We fill this gap with a quantitative definition of ``pragmatic information'', the amount of meaning in a message relevant to a particular decision, a definition that is close to, but not identical with previous definitions. Here, we posit that such a message updates a random variable, $ω$, that informs the decision. The pragmatic information of a single message is then defined as the Kulback-Leibler divergence between the prior and posterior probabilities of $ω$; the pragmatic information of a message ensemble is the expected value of the pragmatic information of the ensemble's component messages. Canonical standard information theory results justify these definitions, most notably that that the pragmatic information of a single message is the expected difference between the shortest binary encoding of $ω$ under the {\it a priori} and {\it a posteriori} distributions. We also show that the average of the pragmatic values of individual messages, when sampled a large number of times from the ensemble, approaches its expected value.\\ Pragmatic information is non-negative and additive for independent decisions and ``pragmatically independent'' messages. Also, pragmatic information is the information analogue of free energy: just as free energy quantifies the part of a system's total energy available to do useful work, so pragmatic information quantifies the information actually used in making a decision.\\ We sketch 3 applications: the single play of a slot machine, a.k.a. a ``one armed bandit'', with an unknown payout probability; a characterization of the rate of biological evolution in the so-called ``quasi-species'' model; and a reformulation of the efficient market hypothesis of finance. We note the importance of the computational capacity of the receiver in each case.

cs.IT↗

arXiv2026

Pragmatic Information Rates, Generalizations of the Kelly Criterion, and Financial Market Efficiency

This paper is part of an ongoing investigation of "pragmatic information", defined in Weinberger (2002) as "the amount of information actually used in making a decision". Because a study of information rates led to the Noiseless and Noisy Coding Theorems, two of the most important results of Shannon's theory, we begin the paper by defining a pragmatic information rate, showing that all of the relevant limits make sense, and interpreting them as the improvement in compression obtained from using the correct distribution of transmitted symbols. The first of two applications of the theory extends the information theoretic analysis of the Kelly Criterion, and its generalization, the horse race, to a series of races where the stochastic process of winning horses, payoffs, and strategies depend on some stationary process, including, but not limited to the history of previous races. If the bettor is receiving messages (side information) about the probability distribution of winners, the doubling rate of the bettor's winnings is bounded by the pragmatic information of the messages. A second application is to the question of market efficiency. An efficient market is, by definition, a market in which the pragmatic information of the "tradable past" with respect to current prices is zero. Under this definition, markets whose returns are characterized by a GARCH(1,1) process cannot be efficient. Finally, a pragmatic informational analogue to Shannon's Noisy Coding Theorem suggests that a cause of market inefficiency is that the underlying fundamentals are changing so fast that the price discovery mechanism simply cannot keep up. This may happen most readily in the run-up to a financial bubble, where investors' willful ignorance degrade the information processing capabilities of the market.

cs.IT↗

arXiv2006

Pragmatic Information and Gaian Development -- First Thoughts

The scientific community believes in the theory of evolution with a passion that rivals that of any religious belief. This passion extends beyond the irrefutable evidence of the fossil record to the familiar claims of survival of the fittest and random mutation. Yet the theory of natural selection has, to the knowledge of the present author, never been tested against the alternative hypothesis that evolution is, in fact, the ongoing development of a single, world-spanning super-organism. Just what kind of evidence would settle this question is not clear, so this paper is intended to begin the discussion. It does so by suggesting how a new, quantitative theory of pragmatic information, first presented in Weinberger (2002), might detect the widespread temporal and inter-species connections that a developmental view of evolution would imply.

nlin.AO↗

arXiv2003

A Theory of Pragmatic Information and Its Application to the Quasispecies Model of Biological Evolution

"Standard" information theory says nothing about the semantic content of information. Nevertheless, applications such as evolutionary theory demand consideration of precisely this aspect of information, a need that has motivated a largely unsuccessful search for a suitable measure of an "amount of meaning". This paper represents an attempt to move beyond this impasse, based on the observation that the meaning of a message can only be understood relative to its receiver. Positing that the semantic value of information is its usefulness in making an informed decision, we define pragmatic information as the information gain in the probability distributions of the receiver's actions, both before and after receipt of a message in some pre-defined ensemble. We then prove rigorously that our definition is the only one that satisfies obvious desiderata, such as the additivity of information from logically independent messages. This definition, when applied to the information "learned" by the time evolution of a process, defies the intuitions of the few previous researchers thinking along these lines by being monotonic in the uncertainty that remains after receipt of the message, but non-monotonic in the Shannon entropy of the input ensemble. It then follows that the pragmatic information of the genetic "messages" in an evolving population is a global Lyapunov function for Eigen's quasi-species model of biological evolution. A concluding section argues that a theory such as ours must explicitly acknowledge purposeful action, or "agency", in such diverse fields as evolutionary theory and finance.

nlin.AO↗