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MathidiaMathematics Academy
Advanced statistics
Lesson 41 of 75
Introduction

Bayesian inference

Objective. Update prior distributions with likelihood evidence.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Bayesian inference updates a prior belief about an unknown parameter into a posterior belief after observing data, using Bayes' theorem: posterior ∝ likelihood × prior. If θ has prior density p(θ) and data x has likelihood p(x|θ), the posterior is p(θ|x) = p(x|θ)p(θ)/p(x), where p(x) is a normalising constant found by integrating the numerator over all θ. Conjugate priors, such as a Beta prior combined with a Binomial likelihood, give a posterior in the same family, simplifying computation. Example: starting with a Beta(1,1) (uniform) prior for a coin's bias and observing 7 heads in 10 tosses gives a Beta(8,4) posterior.

Worked example

Using Bayesian inference with a Beta(1,1) prior and 7 heads in 10 coin tosses, find the posterior distribution's parameters.

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Illustrated card showing 1 … 1
  1. 1The Beta(α,β) prior is conjugate to the Binomial likelihood, so the posterior is also Beta.
  2. 2The posterior parameters update as α' = α + (number of heads) and β' = β + (number of tails).
  3. 3With α = 1, β = 1, 7 heads, and 3 tails: α' = 1 + 7 = 8 and β' = 1 + 3 = 4.
  4. 4So the posterior distribution is Beta(8, 4), reflecting both the prior and observed data.

Answer: Posterior is Beta(8, 4)

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Common mistake. Students often treat the posterior mean as simply the maximum likelihood estimate from the data alone; the posterior combines prior and likelihood, so with an informative prior the posterior mean differs from the raw sample proportion.

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