We’ve seen last time how Bayesian Updating with continuous priors works. As mentioned back then, it isn’t always easy to calculate the total probability of X (p(x)). Today I will introduce the Beta Distribution which makes these calculations easier.

## Beta Distribution

The beta(a,b) distribution is a two parameter distribution on range [0,1] and is therefore ideal to model unknown probabilities:

where c is the normalising constant:

Furthermore the beta distribution is the conjugate prior for Bernoulli, Binomial and Geometric distributions. Conjugate prior means that if our prior is beta our posterior will also be beta.

**Example:** Suppose we have a coin with unknown probability p of heads. We toss the coin 20 times and get 15 heads and 5 tails. What is the posterior PDF?

Hypothesis | Prior | Likelihood | unnormalised Posterior | Posterior |

Total | 1 | 1 |

We can see that the posterior PDF is beta(16,6).