How Can Et Jaynes Probability Theory Help With Priors Selection?

2025-09-03 04:16:19 249

4 Answers

Grayson
Grayson
2025-09-05 17:31:05
I get a little giddy whenever Jaynes comes up because his way of thinking actually makes prior selection feel like crafting a story from what you truly know, not just picking a default. In my copy of 'Probability Theory: The Logic of Science' I underline whole paragraphs that insist priors should reflect symmetries, invariances, and the constraints of real knowledge. Practically that means I start by writing down the facts I have — what units are natural, what quantities are invariant if I relabel my data, and what measurable constraints (like a known average or range) exist.

From there I often use the maximum entropy principle to turn those constraints into a prior: if I only know a mean and a range, MaxEnt gives the least-committal distribution that honors them. If there's a natural symmetry — like a location parameter that shifts without changing the physics — I use uniform priors on that parameter; for scale parameters I look for priors invariant under scaling. I also do sensitivity checks: try a Jeffreys prior, a MaxEnt prior, and a weakly informative hierarchical prior, then compare posterior predictions. Jaynes’ framework is a mindset as much as a toolbox: encode knowledge transparently, respect invariance, and test how much your conclusions hinge on those modeling choices.
Zoe
Zoe
2025-09-06 04:22:46
I tend to tackle priors like I’d prep for a boss fight in a game: gather intel, exploit symmetry, and don’t overcommit to flashy choices. I’ll ask three quick questions: what do I truly know (constraints), what transformations should leave the problem unchanged, and how sensitive is my conclusion to the prior? Jaynes pushes using maximum entropy to encode exactly what you know and nothing more, so if I only know a mean and variance I pick the MaxEnt distribution for those constraints rather than guessing a shape.

A concrete habit I picked up is to try invariance arguments first: if the system is unchanged under shifts, consider a flat prior on location; if it’s unchanged by scale, consider a prior proportional to 1/parameter. But I never stop there — I run posterior predictive checks and priors sensitivity runs. If the posterior barely moves under these reasonable alternatives, I feel safe; if not, I either collect more data or build a hierarchical prior that shares strength across groups. Jaynes made me see priors as explicit encoding of knowledge, not mysterious knobs to tune.
Chase
Chase
2025-09-07 12:11:39
My approach is a little nerdy and a touch formal, but Jaynes’ principles make it concise. I usually start with the desiderata he champions: consistency, invariance under reparameterization where appropriate, and logical coherence with known constraints. That leads to two practical techniques I use in tandem: maximum entropy for encoding soft constraints, and transformation-group arguments for symmetry-based invariance priors.

For example, if I'm modeling the position of an object where shifts in the origin don't matter, I treat the location parameter as having a uniform prior. For a scale parameter (like a standard deviation or a rate), I consider priors that are form-invariant under scaling — often leading to 1/σ-type behavior or Jeffreys’ priors as a starting point. Jaynes steered me away from blind defaults; instead I translate physical or logical symmetries into mathematical demands. I also watch out for improper priors that break normalization and perform posterior predictive checks: simulate data from the prior-predictive, see if simulated datasets look plausible given domain knowledge, and iterate. When I can, I embed vague beliefs into hierarchical models so the data can inform hyperparameters, blending Jaynes’ logic with modern robustness practices.
Knox
Knox
2025-09-07 18:41:38
I like to keep priors honest and minimalistic, and Jaynes’ voice helps me do that. He taught me to convert precise pieces of knowledge into constraints and then use maximum entropy to avoid sneaking in extra assumptions. If all I know is that a parameter is positive, I don’t force a detailed shape — I pick a distribution that reflects positivity and scale-invariance if that’s sensible.

I always complement the theoretical choice with simple checks: prior predictive sampling, sensitivity sweeps with alternative invariant priors, and sometimes a hierarchical layer if borrowing strength is natural. It’s a gentle workflow: encode knowledge, respect symmetries, test robustness, and be ready to collect more data if the prior still decides everything — which usually tells me I need to learn more rather than pretend certainty.
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