289 research outputs found
Disintegration and Bayesian Inversion via String Diagrams
The notions of disintegration and Bayesian inversion are fundamental in
conditional probability theory. They produce channels, as conditional
probabilities, from a joint state, or from an already given channel (in
opposite direction). These notions exist in the literature, in concrete
situations, but are presented here in abstract graphical formulations. The
resulting abstract descriptions are used for proving basic results in
conditional probability theory. The existence of disintegration and Bayesian
inversion is discussed for discrete probability, and also for measure-theoretic
probability --- via standard Borel spaces and via likelihoods. Finally, the
usefulness of disintegration and Bayesian inversion is illustrated in several
examples.Comment: Accepted for publication in Mathematical Structures in Computer
Scienc
Causal inference via string diagram surgery
Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs 'string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a 'twinned' set-up, with two version of the world - one factual and one counterfactual - joined together via exogenous variables that capture the uncertainties at hand
A categorical semantics for causal structure
We present a categorical construction for modelling causal structures within
a general class of process theories that include the theory of classical
probabilistic processes as well as quantum theory. Unlike prior constructions
within categorical quantum mechanics, the objects of this theory encode
fine-grained causal relationships between subsystems and give a new method for
expressing and deriving consequences for a broad class of causal structures. We
show that this framework enables one to define families of processes which are
consistent with arbitrary acyclic causal orderings. In particular, one can
define one-way signalling (a.k.a. semi-causal) processes, non-signalling
processes, and quantum -combs. Furthermore, our framework is general enough
to accommodate recently-proposed generalisations of classical and quantum
theory where processes only need to have a fixed causal ordering locally, but
globally allow indefinite causal ordering.
To illustrate this point, we show that certain processes of this kind, such
as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner,
and a classical three-party example due to Baumeler, Feix, and Wolf are all
instances of a certain family of processes we refer to as in
the appropriate category of higher-order causal processes. After defining these
families of causal structures within our framework, we give derivations of
their operational behaviour using simple, diagrammatic axioms.Comment: Extended version of a LICS 2017 paper with the same titl
The Compositional Structure of Bayesian Inference
Bayes\u27 rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference
Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy
We present a categorical formulation of the cognitive frameworks of
Predictive Processing and Active Inference, expressed in terms of string
diagrams interpreted in a monoidal category with copying and discarding. This
includes diagrammatic accounts of generative models, Bayesian updating,
perception, planning, active inference, and free energy. In particular we
present a diagrammatic derivation of the formula for active inference via free
energy minimisation, and establish a compositionality property for free energy,
allowing free energy to be applied at all levels of an agent's generative
model. Aside from aiming to provide a helpful graphical language for those
familiar with active inference, we conversely hope that this article may
provide a concise formulation and introduction to the framework
- …