Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.

The Core Equation

Where P(A|B) is the posterior probability, P(B|A) is the likelihood, and P(A) is the prior.

Python Simulation

Let's simulate a coin flip experiment. We start with a uniform prior (assuming the coin could be anything from 0% to 100% biased).

import numpy as np

# 1. Define Prior
prior = np.ones(100) / 100 
p_grid = np.linspace(0, 1, 100)

def update(prior, heads, total):
    likelihood = p_grid**heads * (1 - p_grid)**(total - heads)
    posterior = likelihood * prior
    return posterior / posterior.sum()

Summary

As we observe more data, the likelihood dominates the prior, and our posterior distribution converges to the true parameter value.