# Learning in Volatile Environments with the Bayes Factor Surprise

## TL;DR

This paper introduces the Bayes Factor Surprise as a novel measure to improve adaptive learning in volatile environments, leading to new algorithms that outperform existing methods in parameter estimation.

## Contribution

It presents a formalization of surprise as a Bayes Factor, derives three new surprise-based algorithms with simple updates, and demonstrates their effectiveness in non-stationary settings.

## Key findings

- Surprise-modulated algorithms outperform traditional methods in parameter estimation.
- The proposed algorithms have constant complexity and simple update rules.
- Bayes Factor Surprise can be distinguished from Shannon Surprise in physiological predictions.

## Abstract

Surprise-based learning allows agents to rapidly adapt to non-stationary stochastic environments characterized by sudden changes. We show that exact Bayesian inference in a hierarchical model gives rise to a surprise-modulated trade-off between forgetting old observations and integrating them with the new ones. The modulation depends on a probability ratio, which we call "Bayes Factor Surprise", that tests the prior belief against the current belief. We demonstrate that in several existing approximate algorithms the Bayes Factor Surprise modulates the rate of adaptation to new observations. We derive three novel surprised-based algorithms, one in the family of particle filters, one in the family of variational learning, and the other in the family of message passing, that have constant scaling in observation sequence length and particularly simple update dynamics for any distribution in the exponential family. Empirical results show that these surprise-based algorithms estimate parameters better than alternative approximate approaches and reach levels of performance comparable to computationally more expensive algorithms. The Bayes Factor Surprise is related to but different from Shannon Surprise. In two hypothetical experiments, we make testable predictions for physiological indicators that dissociate the Bayes Factor Surprise from Shannon Surprise. The theoretical insight of casting various approaches as surprise-based learning, as well as the proposed online algorithms, may be applied to the analysis of animal and human behavior, and to reinforcement learning in non-stationary environments.