Videos :
How one equation changed the way I think by Julia Galef.
A visual guide to Bayesian thinking by Julia Galef (2015年07月16日).
Simpson's Paradox (14:05) by David Louapre (in French, 2015年04月03日).
The Bayesian Trap (10:36)
by Derek Muller (Veritasium, 2017年04月05日).
Asymmetric Intelligibility in Natural Languages (8:31)
by Josh (2018年06月29日).
Bayesian Statistics (13:47)
Jannah Fry&anbsp; & Matt Parker (2019年03月29日).
In practice, the Bayesian formula (which we shall presently introduce) is often applied to an uncertain hypothesis A and a "test" B for it, relating the a priori probability P(A) to the a posteriori probability P(A|B).
However, both A and B can simply be considered to be events placed on the same footing, which do not play different mathematical rôles:
P(A|B) is read as the probability of "A knowing B". The following holds:
P(A|B) P(B) = P(A,B) = P(B,A) = P(B|A) P(A)
The fact that togetherness is symmetric [i.e., P(A,B) = P(B,A) ] is crucial in the above, which may serve as a proof of the following Bayes' formula of inference, upon which Bayesian statistics is based.
The formula itself is arguably due to Laplace (1812) as Thomas Bayes (1702-1761) didn't bother with such a formal expression. Neither did Richard Price who edited the work of Bayes posthumously (1763).Bayes' Theorem (Bayes' Formula)
This is consistent with a classical description of reality where probabilities are expressed in terms of classical events which could be, among other possibilities, independent or mutually exclusive.
There are probabilistic systems which cannot be described in classical terms (no two events are ever independent or mutually exclusive). The quantum universe (which we live in) is one such system, where the above foundational relation of Bayesian statistics is neither obvious nor true, as experimental violations of Bell's inequality demonstrate.
Bayes' theorem is just true in a self-consistent system where probability is a classical measure satisfying the following axiom of measure theory for the subsets of some universal set E of probability 1.
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Thomas Bayes (c.1701-1761)
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Richard Price (1723-1791)
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Laplace (1749-1827)
Wikipedia :
Bayes' theorem
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Bayesian probability
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Bayesian inference
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Wikipedia : Sequential analysis | Decision theory
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Udny Yule (1871-1951)
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Edward H. Simpson (1922-)
Machine Learning Methods (10:40) by Uwe Aickelin (Computerphile, 2015年09月02日).
Anti-learning (7:50) by Uwe Aickelin (Computerphile, 2015年09月23日).
Wikipedia :
Simpson's Paradox
Video :
Simpson's Paradox (4:39)
by Henry Reich (Minute Physics, 2017年10月24日).
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A Bayesian view of Amazon Resellers
by John D. Cook (2011年09月27日).
Which rating is mathematically better? (12:33)
by Grant Sanderson (2020年03月15日).
In quantum theory, events that can be placed on the same footing correspond to commuting observables. Most pairs of observables do not commute and joint probabilities are strangely defined.
Wikipedia : Quantum Bayesianism | Quantum information
No, it's not. At least not a perfect one. If it was, then it wouldn't be capable of irrational decisions.
Arguably, the biology of the brain involves processes that entail voting and the associated irrational nontransitivy stemming from Condorcet's paradox.
This is not to say, thankfully, that humans cannot consciously revise their beliefs or opinions when presented with new evidence. It merely goes to say that it takes some effort to do so. Just as it takes some effort to practice mathematics and obtain flawless results, in spite of our natural tendencies for intuition, preconceptions and mistakes.
