Bayes' Theorem — Physica

Equivalent forms

P(H\mid D)=\frac{P(D\mid H)P(H)}{\sum_i P(D\mid H_i)P(H_i)}

One line of algebra turns forward probabilities into inference — the engine behind spam filters, medical diagnosis, and modern data analysis.

Symbol	Name	SI unit	Dimension	Range
$P(A\mid B)$	Posterioroutput Updated probability of A given evidence B	1	1	—
$P(B\mid A)$	Likelihood Probability of the evidence if A is true	1	1	0 – 1
$P(A)$	Prior Probability of A before seeing evidence	1	1	0 – 1
$P(B)$	Evidence Total probability of the evidence	1	1	—

Where it holds

Requires well-defined priors and likelihoods. Controversy historically centered on choosing priors; results are exact given the inputs.

Dimensional analysis

P(A|

B) \to

dimensionless (a probability)

P(B|A)P(A

)/P(B) \to

probability

\times

probability/probability

All terms are pure numbers in [0,1]; the ratio is a probability.

\checkmark

Discovery

Thomas Bayes / Pierre-Simon Laplace · 1763

Bayes' essay was published posthumously in 1763 by his friend Richard Price. Laplace independently rediscovered and generalized it, using it to estimate the mass of Saturn and the ratio of male to female births. It languished for a century as 'inverse probability' before becoming central to modern statistics and machine learning.

Research status: stable

Real-world applications

Spam filtering and medical diagnosis
Bayesian parameter estimation in physics (e.g. LIGO, cosmology)
Machine-learning classifiers and Kalman filters

Common misconceptions

P(A| $B) \neq P(B$ |A) — confusing them is the 'prosecutor's fallacy'
A positive result on a 99%-accurate test for a rare disease is usually a false positive
The prior is not optional — 'no prior' quietly assumes a uniform one

Experimental verification

Not a physical law but a theorem of probability; its predictions (e.g., calibrated posterior frequencies) are confirmed in every well-designed inference and A/B test.

Derivation

From the definition of conditional probability, P(A∩

B) = P

(A|

B)P(B) = P(B

|A)P(A).

Equate the two right-hand sides and divide by P(B): P(A|

B) = P(B

|A)P(A)/P(B).

The denominator expands via the law of total probability,

P(B) = \Sigma _i P(B

|A_i)P(A_i), making the posterior a properly normalized distribution over hypotheses.

What if…

What if the prior is 50/50?

The posterior is driven entirely by the likelihood ratio — evidence speaks loudest when you start maximally uncertain.

What if you keep collecting evidence?

Yesterday's posterior becomes today's prior; repeated Bayesian updating converges toward the truth as data accumulate.

1

Rare-disease test

Given ·

prevalence:: 1%
sensitivity:: 99%
false positive:: 5%

Find · P(disease | positive)

Steps

$P(+) = 0.99\times 0.01 + 0.05\times 0.99 = 0.0099 + 0.0495 = 0.0594$
$P(D|+) = 0.0099 / 0.0594$
$\approx 0.167$ — most positives are false

Result ·

\approx 16.7

%

conditional probabilitylaw of total probabilityBinomial Distribution→