arXiv subjects

Simon McGregor

Publications and source records attributed to Simon McGregor.

5 recordsLinked to original sources

arXiv2025

Formalising the intentional stance 2: a coinductive approach

Given a stochastic process with inputs and outputs, how might its behaviour be related to pursuit of a goal? We model this using 'transducers', objects that capture only the external behaviour of a system and not its internal state. A companion paper summarises our results for cognitive scientists; the current paper gives formal definitions and proofs. To formalise the concept of a system that behaves as if it were pursuing a goal, we consider what happens when a transducer (a 'policy') is coupled to another transducer that comes equipped with a success condition (a 'teleo-environment'). An optimal policy is identified with a transducer that behaves as if it were perfectly rational in the pursuit of a goal; our framework also allows us to model constrained rationality. Optimal policies obey a version of Bellman's principle: a policy that's optimal in one time step will again be optimal in the next time step, but with respect to a different teleo-environment (obtained from the original one by a modified version of Bayesian filtering). This property sometimes also applies to the bounded-rational case; we give a sufficient condition. A policy is deterministic if and only if there exists a teleo-environment for which it is uniquely optimal among the set of all policies; we relate this to classical representation theorems from decision theory. This result need not hold in the bounded-rational case; we give an example related to the absent-minded driver problem. The formalism is defined using coinduction, following the style proposed by Czajka.

math.OC↗

arXiv2024

Formalising the intentional stance 1: attributing goals and beliefs to stochastic processes

This article presents a formalism inspired by Dennett's notion of the intentional stance. Whereas Dennett's treatment of these concepts is informal, we aim to provide a more formal analogue. We introduce a framework based on stochastic processes with inputs and outputs, in which we can talk precisely about *interpreting* systems as having *normative-epistemic states*, which combine belief-like and desire-like features. Our framework is based on optimality but nevertheless allows us to model some forms of bounded cognition. One might expect that the systems that can be described in normative-epistemic terms would be some special subset of all systems, but we show that this is not the case: every system admits a (possibly trivial) normative-epistemic interpretation, and those that can be *uniquely specified* by a normative-epistemic description are exactly the deterministic ones. Finally, we show that there is a suitable notion of Bayesian updating for normative-epistemic states, which we call *value-laden filtering*, since it involves both normative and epistemic elements. For unbounded cognition it is always permissible to attribute beliefs that update in this way. This is not always the case for bounded cognition, but we give a sufficient condition under which it is. This paper gives an overview of our framework aimed at cognitive scientists, with a formal mathematical treatment given in a companion paper.

math.OC↗

arXiv2021

Interpreting Dynamical Systems as Bayesian Reasoners

A central concept in active inference is that the internal states of a physical system parametrise probability measures over states of the external world. These can be seen as an agent's beliefs, expressed as a Bayesian prior or posterior. Here we begin the development of a general theory that would tell us when it is appropriate to interpret states as representing beliefs in this way. We focus on the case in which a system can be interpreted as performing either Bayesian filtering or Bayesian inference. We provide formal definitions of what it means for such an interpretation to exist, using techniques from category theory.

cs.AI↗

arXiv2017

The free energy principle for action and perception: A mathematical review

The 'free energy principle' (FEP) has been suggested to provide a unified theory of the brain, integrating data and theory relating to action, perception, and learning. The theory and implementation of the FEP combines insights from Helmholtzian 'perception as inference', machine learning theory, and statistical thermodynamics. Here, we provide a detailed mathematical evaluation of a suggested biologically plausible implementation of the FEP that has been widely used to develop the theory. Our objectives are (i) to describe within a single article the mathematical structure of this implementation of the FEP; (ii) provide a simple but complete agent-based model utilising the FEP; (iii) disclose the assumption structure of this implementation of the FEP to help elucidate its significance for the brain sciences.

q-bio.NC↗

arXiv2015

A Minimal Active Inference Agent

Research on the so-called "free-energy principle'' (FEP) in cognitive neuroscience is becoming increasingly high-profile. To date, introductions to this theory have proved difficult for many readers to follow, but it depends mainly upon two relatively simple ideas: firstly that normative or teleological values can be expressed as probability distributions (active inference), and secondly that approximate Bayesian reasoning can be effectively performed by gradient descent on model parameters (the free-energy principle). The notion of active inference is of great interest for a number of disciplines including cognitive science and artificial intelligence, as well as cognitive neuroscience, and deserves to be more widely known. This paper attempts to provide an accessible introduction to active inference and informational free-energy, for readers from a range of scientific backgrounds. In this work introduce an agent-based model with an agent trying to make predictions about its position in a one-dimensional discretized world using methods from the FEP.

cs.AI↗