Formalised Reasoning about Expectations: Composable, Automated, Speedy, Trustworthy

Automatic Differentiation (AD) systems, like TensorFlow, and probabilistic programming languages (PPLs), like Stan, automate complex computations of derivatives and Bayesian inference tasks.

By stream- lining these computations for non-expert users, these high-level systems have accelerated progress across science and society (e.g. by enabling machine learning).

Yet, the theoretical foundations needed to build a high-level system for composable programming with derivatives and probabilities are missing.

This chasm in our knowledge severely limits the implementation of machine learning techniques, preventing them from reaching their full potential.

Specifically, we do not understand (a) how to perform AD on programs built using probabilistic choices and expected values or (b) how to compose (i.e. combine and integrate) Bayesian inference algorithms.

FoRECAST addresses this chasm by developing programming language theory and tools for flexible, composable, and efficient calculations with derivatives and prob- abilities.

WP 1 develops case studies in collaboration with domain experts, to ensure that FoRECAST creates theory and systems relevant to real-world, complex modelling problems.

WP 2 develops the semantic foundations, algorithms, and formalised correctness proofs for composable AD of probabilistic programs.

WP 3 builds a practical stochastic (i.e. probabilistic) AD system that synthesises these novel gradient estimation techniques. WP 4 establishes theoretical foundations to compose Bayesian inference algorithms in PPLs.

WP 5 implements a user-friendly PPL that facilitates composable Bayesian inference, enabling more flexible modelling for a wider user base.

By mathematically formalising, generalising, optimising, and implementing a next generation PPL, this project will lay a trustworthy foundation upon which probabilistic data analysis applications (e.g. reinforcement learning, proteomics modelling, and paleoclimate reconstructions) can rise to the next level.