Maximum Probability Theorem: A Framework for Probabilistic Learning

Abstract

We present a theoretical framework of probabilistic learning derived by Maximum Probability (MP) Theorem shown in the current paper. In this probabilistic framework, a model is defined as an event in the probability space, and a model or the associated event -- either the true underlying model or the parameterized model -- have a quantified probability measure. This quantification of a model's probability measure is derived by the MP Theorem, in which we have shown that an event's probability measure has an upper-bound given its conditional distribution on an arbitrary random variable. Through this alternative framework, the notion of model parameters is encompassed in the definition of the model or the associated event. Therefore, this framework deviates from the conventional approach of assuming a prior on the model parameters. Instead, the regularizing effects of assuming prior over parameters is seen through maximizing probabilities of models or according to information theory, minimizing the information content of a model. The probability of a model in our framework is invariant to reparameterization and is solely dependent on the model's likelihood function. Also, rather than maximizing the posterior in a conventional Bayesian setting, the objective function in our alternative framework is defined as the probability of set operations (e.g. intersection) on the event of the true underlying model and the event of the model at hand. Our theoretical framework, as a derivation of MP theorem, adds clarity to probabilistic learning through solidifying the definition of probabilistic models, quantifying their probabilities, and providing a visual understanding of objective functions.