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 rigorously justifiable, quantitative definition of ``pragmatic information'', the amount of meaning in a message relevant to a particular decision. We posit that such a message updates a random variable, $\omega$, that informs the decision. The pragmatic information of a single message is then defined as the Kulbach-Leibler divergence between the apriori and aposteriori probabilities of $\omega$; the pragmatic information of a message ensemble is the expected value of the pragmatic information of the ensemble's component messages. We justify these definitions by proving that the pragmatic information of a single message is the expected difference between the shortest binary encoding of $\omega$ under the a priori and a posteriori distributions, and 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.