Breaking down a conditional probability into a marginal probability

Question

I was reading this blog post on using Bayesian statistics to obtain the probability of 2 heads in a row for a coin toss, given that 14 tosses resulted in 10 heads.

I am confused on how the highlighted step in the image was obtained on the page. If anyone could clarify or prove the equality, I'd really appreciate it.

Thanks!

Answer 1

I am confused on how the highlighted step in the image was obtained on the page. If anyone could clarify or prove the equality, I'd really appreciate it.

First observe that

$$\mathbb{P}[HH,p|\text{data}]=\mathbb{P}[HH|p,\text{data}]\cdot\mathbb{P}[p|\text{data}]$$

now considering the assumption they made of conditional independence (w.r.t. the parameter $p$) of the observations you get

$$\mathbb{P}[HH,p|\text{data}]=\mathbb{P}[HH|p]\cdot\mathbb{P}[p|\text{data}]$$

At this point, to eliminate $p$ in the left member you have to integrate both members in $dp$ obtaining

$$ \bbox[5px,border:2px solid black] { \mathbb{P}[HH|\text{data}]=\int_0^1\mathbb{P}[HH|p]\cdot\mathbb{P}[p|\text{data}]dp \ } $$

Observe that inside the integral you have

$$\underbrace{\mathbb{P}[HH|p]}_{=\text{model}}\cdot\underbrace{\mathbb{P}[p|\text{data}]}_{=\text{Posterior Distribution}}$$