# Statistical Epistemic Logic

## TL;DR

This paper introduces a novel modal logic called statistical epistemic logic that models statistical knowledge, including non-deterministic inputs and significance, with applications in secrecy, hypothesis testing, and privacy.

## Contribution

It presents the first semantics for modal logic capable of expressing statistical knowledge based on probabilistic and stochastic information.

## Key findings

- Defines a Kripke model for probability distributions and stochastic assignments.
- Provides a semantics for expressing statistical secrecy and significance.
- Formalizes statistical hypothesis testing and differential privacy using the logic.

## Abstract

We introduce a modal logic for describing statistical knowledge, which we call statistical epistemic logic. We propose a Kripke model dealing with probability distributions and stochastic assignments, and show a stochastic semantics for the logic. To our knowledge, this is the first semantics for modal logic that can express the statistical knowledge dependent on non-deterministic inputs and the statistical significance of observed results. By using statistical epistemic logic, we express a notion of statistical secrecy with a confidence level. We also show that this logic is useful to formalize statistical hypothesis testing and differential privacy in a simple and abstract manner.