A Notational Framework for Contextual Inference in Scientific Modelling
A Notational Framework for Contextual Inference in Scientific Modelling
By Stefaan Vossen
25/02/2026
Abstract
Scientific inference is commonly expressed as the estimation of latent variables from observed data. In practice, however, the interpretation of observations depends on contextual information such as measurement conditions, modelling assumptions, coordinate conventions, prior knowledge, and task objectives.
This paper introduces the Dot operator ⊙ as a compact notation for contextual inference. The operator represents the mapping from observations and contextual metadata to posterior beliefs over latent variables: D ⊙ M(ψ) = p(X ∣ D, M(ψ)).
The contribution of this paper is conceptual rather than mathematical. The operator does not introduce new probability theory. Instead it provides a symbolic representation of the contextual conditioning step underlying Bayesian inference and probabilistic modelling.
We show that the Dot operator corresponds to Bayesian inversion of a generative model and can be interpreted formally as a posterior Markov kernel within the framework of categorical probability. This situates the notation within established mathematical structures used in probabilistic programming and stochastic process theory.
The framework clarifies the role of contextual information in scientific modelling and provides a compact representation of contextual inference across statistics, machine learning, and observational science.
1 Introduction
Many scientific disciplines rely on inference over latent variables. Observed data are interpreted through models that specify how observations arise from hidden states.
In statistical notation inference is typically written as:
p(X ∣ D)
where X denotes latent variables and D denotes observations.
In practice observations rarely possess meaning independent of contextual structure. Interpretation typically depends on:
-measurement calibration
-experimental design
-coordinate conventions
-model assumptions
-prior knowledge
-task objectives.
Modern statistical frameworks already incorporate contextual information through priors, covariates, hierarchical models, and generative assumptions. Machine learning models similarly condition predictions on contextual features or embeddings.
Despite this widespread use of contextual conditioning, the inferential step integrating observations and contextual information is rarely represented explicitly in symbolic form. This paper introduces a simple notation for that step. The Dot operator ⊙ denotes contextual inference in which observations are interpreted relative to contextual metadata.
The aim is conceptual clarification. The notation emphasises a structural feature shared by many probabilistic modelling frameworks.
2 Data and metadata as task dependent roles
Analytical workflows typically distinguish between two forms of information
-data
-metadata.
Data are observations relevant to the phenomenon under study. Metadata describe contextual information about how those observations were generated. However the distinction between these categories is not fixed, examples include:
• time acting as metadata when interpreting measurements but as primary data in time series analysis
• calibration parameters functioning as metadata in experiments but as primary variables in instrumentation research
• patient context variables serving as metadata in diagnostic systems but as outcomes in epidemiological studies.
This motivates the following Principle of contextual interpretation:
The distinction between data and metadata is task dependent. Variables function as data or metadata depending on the modelling objective and contextual framework used to interpret observations.
Information therefore has no intrinsic epistemic role. Its role emerges from the modelling task.
3 The Dot operator
Let:
D ∈ 𝒟 denote observations
ψ ∈ Ψ denote observer or model state
M(ψ) ∈ ℳ denote metadata derived from that state
X ∈ 𝒳 denote latent variables.
In Bayesian modelling inference typically takes the form
p(X ∣ D, M).
The Dot operator ⊙ provides a compact representation of this contextual inference step.
Definition: D ⊙ M(ψ) = p(X ∣ D, M(ψ)).
The operator therefore has type ⊙ ∶ 𝒟 × ℳ → 𝒫(𝒳).
The operator does not transform observations themselves. Instead it represents the inference step mapping observations and contextual metadata to posterior beliefs.
4 Generative structure
Probabilistic inference typically arises from a generative model.
Define π ∶ ℳ → 𝒫(𝒳) as a prior distribution over latent variables conditioned on metadata.
Define g ∶ 𝒳 × ℳ → 𝒫(𝒟) as a likelihood model generating observations.
These components define a joint distribution p(X, D ∣ M).
Posterior inference corresponds to inversion of this joint distribution.
post ∶ 𝒟 × ℳ → 𝒫(𝒳).
The Dot operator corresponds to this posterior mapping ⊙ ≡ post.
5 Theorem: equivalence with Bayesian conditioning
The Dot operator introduces no new probability theory.
Theorem
Given a generative model
π ∶ ℳ → 𝒫(𝒳)
g ∶ 𝒳 × ℳ → 𝒫(𝒟)
the contextual inference operator satisfies
D ⊙ M = p(X ∣ D, M).
Proof
From Bayes rule
p(X ∣ D, M) = p(D ∣ X, M) p(X ∣ M) / p(D ∣ M).
Substituting the generative components yields
p(X ∣ D, M) ∝ g(X, M) π(M).
Therefore the Dot operator corresponds to posterior inference under the generative model.
6 Categorical probability and Markov kernel interpretation
The Dot operator can be expressed naturally within the framework of categorical probability and Markov kernels. This section shows that the operator corresponds to Bayesian inversion of a generative model in the sense used in categorical probability theory.
The purpose of this section is not to introduce new mathematical results but to situate the operator within an existing formal framework.
6.1 Markov kernels
Let 𝒳 and 𝒴 be measurable spaces.
A Markov kernel
k ∶ 𝒳 → 𝒫(𝒴)
assigns to each x ∈ 𝒳 a probability distribution over 𝒴.
Sequential composition of kernels is defined by marginalisation
(h ∘ k)(z ∣ x) = ∫ h(z ∣ y) k(y ∣ x) dy.
This structure forms the basis of probabilistic computation and stochastic process theory.
6.2 Generative models as kernel composition
Generative probabilistic models can be expressed using Markov kernels.
Let
π ∶ ℳ → 𝒫(𝒳)
be a prior kernel.
Let
g ∶ 𝒳 × ℳ → 𝒫(𝒟)
be a likelihood kernel.
The generative structure becomes
ℳ → 𝒳 → 𝒟.
These kernels induce a joint distribution
p(X, D ∣ M).
6.3 Bayesian inversion
Inference corresponds to inversion of the generative kernel.
Given observations D and metadata M the posterior kernel is
post ∶ 𝒟 × ℳ → 𝒫(𝒳)
defined by
post(X ∣ D, M) = p(X ∣ D, M).
This operation is known as Bayesian inversion or disintegration.
6.4 Dot operator as posterior kernel
Within this framework the Dot operator corresponds to the posterior kernel
D ⊙ M = post(X ∣ D, M).
Thus the operator
⊙ ∶ 𝒟 × ℳ → 𝒫(𝒳)
implements Bayesian inversion of the generative kernel.
6.5 Relation to Markov categories
Markov categories provide a categorical structure for probabilistic systems in which morphisms represent stochastic maps.
Within this framework
objects correspond to measurable spaces
morphisms correspond to Markov kernels.
The generative process
ℳ → 𝒳 → 𝒟
is a composition of stochastic morphisms.
Posterior inference corresponds to constructing a kernel
𝒟 × ℳ → 𝒳
satisfying the Bayesian inversion property.
The Dot operator can therefore be interpreted as a stochastic morphism implementing Bayesian inversion.
7 Worked example
Consider a temperature sensor measuring environmental temperature.
Observed data
D = sensor reading.
Latent variable
X = true temperature.
Metadata
M = calibration offset and noise model.
Inference takes the form
p(X ∣ D, M).
Using the Dot operator
D ⊙ M = p(X ∣ D, M).
Suppose the likelihood model is Gaussian
p(D ∣ X, M) = 𝒩(X + c, σ²).
The posterior estimate becomes
D ⊙ M ∝ 𝒩(D − c, σ²).
This example illustrates how contextual metadata influence the interpretation of observations.
8 Recursive inference
Inference systems often update internal state as new observations arrive.
ψₜ₊₁ = Update(ψₜ, Dₜ).
Metadata evolve accordingly
Mₜ = M(ψₜ).
Contextual inference therefore becomes
Dₜ ⊙ Mₜ = p(Xₜ ∣ Dₜ, Mₜ).
This recursive structure appears in
Bayesian filtering
reinforcement learning
predictive processing models.
9 Contextual inference in scientific modelling
Many scientific quantities are not directly observed but inferred from data.
Examples include
cosmological parameters inferred from astronomical surveys
particle properties inferred from detector signals
spacetime geometry inferred from gravitational observations.
In such cases inference takes the form
D ⊙ M(ψ) = p(X ∣ D, M(ψ)).
Here
X represents latent physical quantities
M represents modelling assumptions and calibration information.
This interpretation does not modify the equations of physical theories. Instead it clarifies the inferential structure through which models are applied to observations.
10 Discussion
Many influential symbolic systems reorganised existing theory without introducing new mathematics.
Examples include
Leibniz differential notation
Dirac bra ket notation
Feynman diagrams
category theoretic diagrams.
The Dot operator is proposed in a similar spirit. Its purpose is to provide a compact symbolic representation of contextual inference.
11 Conclusion
The distinction between data and metadata is task dependent.
The Dot operator
D ⊙ M(ψ) = p(X ∣ D, M(ψ))
provides a compact representation of contextual inference.
Formally the operator corresponds to Bayesian inversion
⊙ ∶ 𝒟 × ℳ → 𝒫(𝒳)
and can be interpreted as a posterior Markov kernel within categorical probability.
References
Cox, R. T. (1946). Probability, frequency, and reasonable expectation. American Journal of Physics.
Jaynes, E. T. (2003). Probability theory: the logic of science. Cambridge University Press.
Pearl, J. (1988). Probabilistic reasoning in intelligent systems. Morgan Kaufmann.
Friston, K. (2010). The free energy principle. Nature Reviews Neuroscience.
Fong, B., Jacobs, B. (2019). A categorical approach to probability theory.
Fritz, T. (2020). A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statistics. Advances in Mathematics.
Mac Lane, S. (1998). Categories for the working mathematician. Springer.
Glossary
Data D
Observed measurements produced by an experimental or observational process.
Metadata M
Contextual information describing measurement conditions, modelling assumptions, or prior knowledge.
Observer state ψ
Internal state of a model or inference system from which metadata may be derived.
Latent variables X
Unobserved quantities inferred from observations.
Markov kernel
A mapping assigning a probability distribution to each element of a measurable space.
Bayesian inversion
Derivation of posterior distributions from generative models.
Dot operator ⊙
Symbolic representation of contextual inference defined by
D ⊙ M(ψ) = p(X ∣ D, M(ψ)).
Appendix A
Notation
𝒟 space of observations
ℳ space of metadata
𝒳 space of latent variables
Ψ space of observer states
𝒫(𝒳) space of probability distributions over 𝒳
Appendix B
Bayesian inversion
The posterior distribution
p(X ∣ D, M) is obtained from the joint distribution
p(X, D ∣ M) via Bayes rule
p(X ∣ D, M) = p(D ∣ X, M) p(X ∣ M) / p(D ∣ M).
This corresponds to the Dot operator
D ⊙ M = p(X ∣ D, M).
