Foundations of Induction - of Marcus Hutter

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Marcus Hutter - 28 - Foundations of InductionSome other Properties of M• Problem of zero prior / confirmation of universal hypotheses:{ ≡ 0 in Bayes-Laplace modelP[All ravens black|n black ravens] fast−→ 1 for universal prior wθU• Reparametrization and regrouping invariance: wθU = 2−K(θ) alwaysexists and is invariant w.r.t. all computable reparametrizations f.(Jeffrey prior only w.r.t. bijections, and does not always exist)• The Problem of Old Evidence: No risk of biasing the prior towardspast data, since wθ U is fixed and independent of model class M.• The Problem of New Theories: Updating of M is not necessary,since universal class M U includes already all.• M predicts better than all other mixture predictors based on any(continuous or discrete) model class and prior, even innon-computable environments.
Marcus Hutter - 29 - Foundations of InductionMore Stuff / Critique / Problems• Prior knowledge y can be incorporated by using “subjective” priorw U ν|y = 2−K(ν|y) or by prefixing observation x by y.• Additive/multiplicative constant fudges and U-dependence is often(but not always) harmless.• Incomputability: K and M can serve as “gold standards” whichpractitioners should aim at, but have to be (crudely) approximatedin practice (MDL [Ris89], MML [Wal05], LZW [LZ76], CTW [WSTT95],NCD [CV05]).
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Foundations of Induction - of Marcus Hutter

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