# How to quantify strength of beta prior?

How would one interpret the strength of prior belief associated with a parameter with a prior beta(10,8) distribution, compared to a beta(0,0) prior, (with data from a binomial distribution)? Some intuition and calculations to support your answer would be really appreciated.

• I took the liberty to edit your title to something more informative. Feel free to revert the change it edit it if you feel that it is inaccurate. – Tim Mar 12 '18 at 5:47

## 1 Answer

Consider the standard Bayesian conjugate model:

$$\theta \sim \text{Beta}(\alpha, \beta),$$ $$X | \theta \sim \text{Bin}(n, \theta).$$

The posterior distribution for this model is:

$$\theta | x \sim \text{Beta}(\alpha + x, \beta + n-x).$$

In this model, the value $n_0 \equiv \alpha + \beta$ in the prior is a measure of prior strength that is sometimes called the number of pseudo data points. Observation of the count value $x$ from $n$ trials adds to this to the posterior strength $n_0 + n$, so this is like you had $n_0$ pseudo data points and then added $n$ actual data points.