The P–T Probability Framework for Semantic Communication, Falsification, Confirmation, and Bayesian Reasoning

Abstract

Many researchers want to unify probability and logic by defining logical probability or probabilistic logic reasonably. This paper tries to unify statistics and logic so that we can use both statistical probability and logical probability at the same time. For this purpose, this paper proposes the P–T probability framework, which is assembled with Shannon’s statistical probability framework for communication, Kolmogorov’s probability axioms for logical probability, and Zadeh’s membership functions used as truth functions. Two kinds of probabilities are connected by an extended Bayes’ theorem, with which we can convert a likelihood function and a truth function from one to another. Hence, we can train truth functions (in logic) by sampling distributions (in statistics). This probability framework was developed in the author’s long-term studies on semantic information, statistical learning, and color vision. This paper first proposes the P–T probability framework and explains different probabilities in it by its applications to semantic information theory. Then, this framework and the semantic information methods are applied to statistical learning, statistical mechanics, hypothesis evaluation (including falsification), confirmation, and Bayesian reasoning. Theoretical applications illustrate the reasonability and practicability of this framework. This framework is helpful for interpretable AI. To interpret neural networks, we need further study.

Information in Operation: Probability

“…If subjective predictions always accord with facts, the two information accounts are the same. As presented by Chenguang Lu in two articles in a Special Issue of Philosophies, the first approach deals with how information is acquired, a field often referred to as Informatics [5]. The second, different approach of Information Theory deals with information measurement.…”

Using the Semantic Information G Measure to Explain and Extend Rate-Distortion Functions and Maximum Entropy Distributions

“…These conclusions accord with Popper's thoughts [37] (p. 294). For this reason, I(xi; θj) is also explained as the verisimilitude between yj and xi [32]. We can also use the above formula to measure sensory information, for which T(θj|x) is the confusion probability function of xj with x or the discrimination function of xj [31].…”

“…To increase G = I(X; Yθ), we need to fix T(θj|x) and optimize control to improve P(x|y). The R(G) function tells us that we can improve P(y|x) with Equation (15) and increase G by amplifying s. In [32], the author improperly uses a different information formula (Equation (24) in [32]) for the above purpose. That formula seemingly only fits cases where the control results are continuous distributions.…”

“…The G theory uses the P-T probability framework [32], which consists of both statistical probability (denoted by P) and logical probability (by T). The P-T probability framework and the G theory have been applied to several areas, such as data compression related to visual discrimination [30], machine learning [31], Bayesian confirmation [33], and the philosophy of science [32].…”

“…For overcoming the rate-distortion function's disadvantages, this paper sets up a transformation relation between the truth function and the distortion function and uses the truth function to replace the distortion function for the constraint condition to obtain the rate-truth function. Since a truth function can also be explained as a membership function, similarity function, confusion probability function, or DCF [32], it can be used as a learning function. It is often expressed as an NEF or Logistic function.…”