Bayesian Inference: Grid Approximation

This interactive demo illustrates Bayesian inference using grid approximation. Imagine we're surveying people about whether they drink coffee. We want to estimate the true proportion of coffee drinkers in the population.

How It Works

1. Prior: Choose your prior belief (uniform or centered on a specific value)
2. Data: Each "Yes" or "No" click represents a survey response
3. Posterior: The distribution updates based on Bayes' theorem

Original concept from Probabilistic Machine Learning course (Summer Term 2025)
University of Tübingen, taught by Professor Philipp Hennig

Controls

Posterior Distribution

Understanding the Math

Bayes' Theorem

Where:

• is the prior — our initial belief about
• is the likelihood — probability of seeing this data given
• is the posterior — our updated belief after seeing data

Likelihood Function

For binomial data (yes/no responses), the likelihood is:

where is the number of "yes" responses and is total responses.

Prior Distributions

Uniform Prior: All values of are equally likely:

Gaussian Prior: Centered at with standard deviation :

Grid Approximation

Instead of solving analytically, we:

1. Create a fine grid of possible values from 0 to 1
2. Compute the likelihood at each grid point
3. Multiply by the prior
4. Normalize so the total probability sums to 1

This gives us a discrete approximation of the true posterior distribution!