# Bayesian Inference V – Bayesian Updating Continuous Priors

We already know how to do Bayesian Updating with discrete priors. Today we will learn how to do Bayesian Updating with continuous priors.

## Continous Priors

To do Bayesian Updating with continuous priors but with discrete data – we will look at the case that both is discrete next time – we just change sums to integrals and PMFs to PDFs.

We often denote the hypothesis $\theta$ “theta” where theta is a range and not a discrete hypothesis. The hypothesis is then often written as: $H:\;\theta \pm \frac{ d\theta }{ 2 }$, what means that the hypothesis is in a small interval of width $d\theta$ around $\theta$. That is because the probability is given by $f(\theta)d\theta$

If we apply all of the above mentioned things we get a Bayesian Updating table that looks like this:

 Hypothesis Prior Likelihood unnormalised Posterior Posterior $\theta \pm \frac{ d\theta }{ 2 }$ $f(\theta)d\theta$ $p(x|\theta)$ $p(x|\theta)f(\theta)d\theta$ $\frac{ 1 }{ T }p(x|\theta)f(\theta)d\theta$ Total $\int_{ a }^{ b }{ f(\theta)d\theta }=1$ $T=\int_{ a }^{ b }{p(x|\theta)f(\theta)d\theta }$ 1

$p(x)$ which is the total of the unnormalised posterior column can be hard to calculate and we often use a computer to do so, in case we don’t have a conjugate prior. More about conjugate priors soon.

Example: Suppose we have a coin with flat prior and unknown probability p of heads. We flip the coin and get heads. What is the posterior PDF?

Answer: A flat prior means that $f(\theta)=1$ on the interval [0,1] which mens that all hypotheses are equally likely and that the hypotheses are between 0 and 1. Furthermore we let x=1 mean that the outcome was heads.

The definition of our hypothesis tells us that $p(x|\theta)=\theta$ because our probability of getting heads is our hypothesis. We then have the following Bayesian Updating table:

 Hypothesis Prior Likelihood unnormalised Posterior Posterior $\theta \pm \frac{ d\theta }{ 2 }$ $1d\theta$ $\theta$ $\theta d\theta$ $2\theta d\theta$ Total 1 $T=\int_{ 0 }^{ 1 }{ \theta d\theta } =\frac{ 1 }{ 2 }$ 1

Our posterior PDF after seeing one heads is then: $f(\theta|x)=2\theta$

We can see that Bayesian Updating with continuous priors doesn’t differ much from Bayesian Updating with discrete priors.