Bayes, precision and belief updating

Bayesian updating: how beliefs change when evidence arrives

Bayesian updating is a formal way to combine what we already believe with new evidence. The key idea is not just whether the evidence points one way or another, but how precise each source of information is.

1. The basic idea

A prior belief describes what we expected before seeing the data. A likelihood describes how compatible the new evidence is with different possible hidden states. The posterior is the updated belief after combining both.

$$p(\theta \mid y) = \frac{p(y \mid \theta)p(\theta)}{p(y)}$$

In this page, we use Gaussian beliefs because they are visually intuitive: each curve has a centre and a width. Narrow curves mean high precision. Wide curves mean low precision.

2. Interactive Bayesian updating

Move the sliders to see how prior belief and sensory evidence combine. Notice that a precise prior resists change, while precise evidence pulls the posterior strongly.

Prior mean -1.2 Prior uncertainty 1.1 Evidence value 1.4 Evidence uncertainty 0.65

Posterior mean 0.66

Posterior uncertainty 0.56

Interpretation Evidence is more precise than the prior, so the posterior moves toward the evidence.

3. Precision weighting

For Gaussian beliefs, updating can be written in terms of precision. Precision is the inverse of variance:

$$\pi = \frac{1}{\sigma^2}$$

The posterior mean is a precision-weighted compromise between the prior and the evidence:

$$\mu_{\text{post}} = \frac{\pi_{\text{prior}}\mu_{\text{prior}} + \pi_{\text{evidence}}\mu_{\text{evidence}}} {\pi_{\text{prior}} + \pi_{\text{evidence}}}$$

This is one of the most useful bridges into computational neuroscience. In predictive coding, the brain is often described as updating beliefs in proportion to precision-weighted prediction errors.

4. Why this matters for our lab

Many of our models estimate hidden causes from noisy biological data. EEG and MEG signals are indirect measurements of neural activity and psychiatric symptoms are indirect measurements of underlying mechanisms. Bayesian updating gives us a principled way to combine prior assumptions, empirical data and uncertainty.

In Dynamic Causal Modelling and Parametric Empirical Bayes, this logic appears repeatedly: subject-level parameters are estimated with uncertainty, group-level effects update beliefs about those parameters and posterior probabilities express how confident we are that a mechanism changed.