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Bayes' theorem and inference

Updated 11 Oct 2026

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Bayes' theorem
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  $P(H\mid E)=\frac{P(E\mid H)P(H)}{P(E)}$ , when $P(E)>0$ .
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  for exhaustive, mutually exclusive hypotheses $H_i$ :
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    $P(E)=\sum_i P(E\mid H_i)P(H_i)$ .
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  Prior -> $P(H)$ , before observing evidence.
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  Likelihood -> $P(E\mid H)$ , evidence probability assuming $H$ .
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  Posterior -> $P(H\mid E)$ , updated probability after evidence.
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  Common mistake -> reverse the conditional.
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Law of large numbers
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  under its assumptions, empirical averages approach their expected value as the number of trials grows.
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  it does not guarantee that short sequences balance out immediately.
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Confidence intervals
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  A Confidence interval describes long-run coverage under its assumptions
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  $95\%$ procedure -> captures the target parameter in about $95\%$ of repeated samples.
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  It does not mean $95\%$ of individual observations lie inside the interval.
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  It does not automatically correct selection bias or measurement error.

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