Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: semantics and automated reasoning with theorem proving

TL;DR

The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards the vision to tackle the challenges of predicative probabilistic programming, based on Hehner's predicative probabilistic programming.

Abstract

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards our vision to tackle these challenges. Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving. We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.

Figures (5)

• Figure 1: Termination probability over $t'$ for dice.
• Figure 2: Complete lattices: left $\left(\textit{ureal}, \mathrel{\mathstrut{\leq}}\right)$ and right $\left(S \mathbin{\mathstrut{\rightarrow}} \textit{ureal}, \mathrel{\mathstrut{\leq}}\right)$. We use $\left\{{\frac{3}{4}}, {\frac{1}{4}}\right\}$ denotes a function $\left\{s_1 \mathbin{\mathstrut{\mapsto}} {\frac{3}{4}}, s_2 \mathbin{\mathstrut{\mapsto}} {\frac{1}{4}}\right\}$ whose domain contains two elements $s_1$ and $s_2$ and their corresponding probabilities are ${\frac{3}{4}}$ and ${\frac{1}{4}}$ respectively. $\mathring{\frac{1}{4}}$ denotes a constant function which maps every element in its domain to ${\frac{1}{4}}$. Dashed box: subdistributions; normal: distributions; thick: superdistributions.
• Figure 3: Termination probability over iterations from bottom and top for coin flip.
• Figure 4: Illustration of limits of an increasing chain for various states.
• Figure 5: The update of the robot's belief at different positions after three measurements and two moves with a prior.

Theorems & Definitions (157)

• definition 3 . 1: Constructs of sequential programs
• definition 3 . 2: Monotone and anti-monotone
• definition 3 . 3: Chains
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3: Limit as supremum and infimum
• definition 3 . 4: Limit of a sequence
• Theorem 3.4: Knaster–Tarski fixed-point theorem
• definition 3 . 5: Scott continuity Abramsky1995
• Theorem 3.5: Monotonicity