PRB-02

Bayes

Invert the conditional.

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ProbabilityBayes

Governing equation

P(A|B)=P(B|A)P(A)/P(B)

$P(A) \in \left[0.001, 0.5\right]$

where

P(B|A): P(B|A) (—)
P(A): P(A) (—)
P(B): P(B) (—)
P(A|B): P(A|B) (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

P(A|B)=P(B|A)P(A)/P(B)

$P(A) \in \left[0.001, 0.5\right]$

$P(A) = 0.001$ $P(A|B) = 0.018$

Along the curve

②

Domain of the variable — blue

P(A)_{\min} = 0.001,\quad P(A)_{\max} = 0.5

$P(A) \in \left[0.001, 0.5\right]$

$P(A) = 0.001$ $P(A) = 0.001$

Along the curve

③

Response curve — green

P(A|B)=P(B|A)P(A)/P(B)

$P(A) \in \left[0.001, 0.5\right]$

$P(A) = 0.001$ $P(A|B) = 0.018$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}P(A|B)}{\mathrm{d}P(A)}

$P(A) \in \left[0.001, 0.5\right]$

$P(A) = 0.004$ $P(A|B)^{\prime} = 18$

Along the curve

Lecture brief

Historical brief

Bernoulli trials, Poisson, Gauss, Bayes and Shannon entropy turned chance into a calculus of belief and noise. The lab computes likelihood, interval and information in SI-free counts. This sheet (PRB-02 — Bayes) is the form associated with Bayes. Working symbols: $P(B|A)$ , $P(A)$ , $P(B)$ $\rightarrow$ $P(A|B)$ . Invert the conditional.

Purpose

Purpose: compute

P(A|B)

from

P(B|A)

P(A)

P(B)

in Probability & statistics via

P(A|B)=P(B|A)P(A)/P(B)

Invert the conditional. Use it when a real probability & statistics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

P(B|A) = 0.900\,\mathrm{—}

P(A) = 0.010\,\mathrm{—}

P(B) = 0.050\,\mathrm{—}

, the governing relation

P(A|B)=P(B|A)P(A)/P(B)

yields

P(A|B) = 0.1800\,\mathrm{—}

. Invert the conditional. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

P(B|A) P(B|A)0.9 —P(A) P(A)0.01 —P(B) P(B)0.05 —

Outputs

P(A|B) P(A|B)0.1800 —

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Narration of this film

Invert the conditional.

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