Mathematical Notes for Bayesian Updates and Posterior Diagnostics in skpro

[arnavk23]

Related PRs

• [ENH] Add BaseBayesianRegressor for Bayesian model interface #802
• [ENH] BayesianConjugateLinearRegressor online update in precision form #771
• [ENH] Improve Bayesian tutorial notebooks #791
• [ENH] Add naive baseline with kernel, parametric and bounding estimators #786
• [ENH] Add TemporalNormal distribution for time-dependent Gaussian #785
• [ENH] Implement outlier detection based on probabilistic regressors #777

Explicit PR-To-Topic Mapping

PR	Main contribution	How it maps to notes below
#771	BayesianConjugateLinearRegressor online update in precision form	Directly maps to Sections 1, 2, 4, 5, 6, and 8 (sequential update, conjugate posterior, diagnostics, precision update, numerical stability).
#802	BaseBayesianRegressor interface and posterior/predictive workflow	Maps to Sections 1, 3, and 7 (posterior update contract at interface level, predictive posterior representation, MC/sample-based posterior handling).
#791	Bayesian tutorial improvements	Indirect/documentation mapping: clarifies notation and presentation used across Sections 1 to 3 and 8.
#785	TemporalNormal distribution	Related probabilistic context, but not a direct source of conjugate linear posterior update formulas in this note.
#786	Naive baseline/kernel/parametric/bounding estimators	Related probabilistic estimator tooling, but not a direct source of Bayesian conjugate update derivations here.
#777	Outlier detection from probabilistic regressors	Uses predictive distributions, but not a direct source of posterior update equations in this note.

Was finding it hard to write completely in this, so wrote in latex and putting the pdf here for your reading @fkiraly
bayesian_pr_notes.pdf.

I think as we were clearly things out today I got a bit flustered and started to go the wrong direction, majorly I use precision as inverse of variance matrix but in the questioning felt as if you were talking about a different matrix which I hadn't read up on so answered incorrectly. And by computational I was meaning doing the statistical calculation on a computer not getting close to the point (using float). I apologise for today and will try to compose myself better in our future meetings.

Also I plan to keep this issue active throughout my work on skpro as I think this can be a good discussing place for both of us if there are any issues/major questions/explanations to give.

I thank you for your time and consideration.

[arnavk23]

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