[D] A notation for contextual inference in probabilistic models

Hello everyone,

I am looking for critical feedback on an idea that could look somewhat redundant but has the potential to clarify how modelling context and observed data interact in probabilistic inference.

In many scientific models, inference is formally expressed as conditioning on observed data, yet in practice the interpretation of observations also depends on contextual information such as modelling assumptions, calibration parameters, and prior knowledge. This paper introduces a simple notation for representing that contextual inference step explicitly, expressing the mapping from observations and modelling context to posterior beliefs as:

D ⊙ M(ψ) = p(X ∣ D, M(ψ)).

I wrote this short conceptual paper proposing a simple notation for contextual inference in probabilistic modelling and I would be interested in feedback from people working in ML theory or probabilistic modelling.

Post:

The linked short paper proposes a notational framework for representing contextual inference in scientific modelling.

In many modelling pipelines we write inference as

p(X ∣ D)

but in practice predictions depend not only on the data but also on contextual structure such as

• calibration parameters
• modelling assumptions
• task objectives
• prior information.

The paper introduces a compact notation:

D ⊙ M(ψ)

to represent the step where observations are interpreted relative to contextual metadata.

Formally this is just standard Bayesian conditioning

D ⊙ M(ψ) = p(X ∣ D, M(ψ))

so the goal is not to introduce new probability theory, but to make the contextual conditioning step explicit. The motivation for this notation is to make explicit the structural role of context in probabilistic inference, clarifying how observations are interpreted relative to modelling assumptions and potentially improving the transparency and composability of scientific models.

The paper connects this notation to

• generative models
• Bayesian inversion
• Markov kernels
• categorical probability.

In categorical terms the operator corresponds to the posterior kernel obtained by disintegration of a generative model.

The motivation is mainly structural. Modern ML systems combine observations with contextual information in increasingly complex ways, but that integration step is rarely represented explicitly at the level of notation.

I would be interested in feedback on whether something equivalent to this notation already exist in categorical probability or probabilistic programming frameworks and either:

• this perspective already exists in ML literature
• the notation is redundant
• something similar appears in probabilistic programming frameworks or

• it is novel and possibly useful

The paper is short and intended as a conceptual methods note but, by extension in such fields as statistics, machine learning, probabilistic programming, and scientific modelling, the notation may help clarify how contextual information enters inference and clarify how observations are interpreted within modelling frameworks.

Thank you for your time and attention,

Stefaan

https://www.dottheory.co.uk/paper/a-notational-framework-for-contextual-inference-in-scientific-modelling