what is the step by step procedure of updating a posterior probability with new data coming in

Question

I have a question about how to update posterior probability sequentially when new data comes in sequentially by say $x_1$, then $x_2$, then $x_3$,.... I understand this form when the first data $x_1$ comes in (this very first updating is based on the Baye's formula):

$p(\theta |x_1) = \frac{p(x_1|\theta) * p(\theta)}{\int p(x_1|\theta) *p(\theta) d\theta}$

But how about when $x_2$ is coming in next now? what would be the updating looks like now? could someone gives me step by step procedures?

because I see a form like this when the sixth data ($x_6$) is included:

$p(\theta|x_1,x_2,...,x_6)=\frac{p(x_2,x_3,...,x_6|\theta) *p(\theta|x_1)}{\int p(x_2,x_3,...,x_6|\theta)*p(\theta|x_1)d\theta}$

But I couldn't understand how it arrives.

Could someone shows me how to go from the very first updating, to the form of $p(\theta|x_1,x_2)$ ? because I think once I know how to update the posterior from $x_1$ to $x_2$, then I would be able to figure out how to reach the posterior when the sixth data $x_6$ is included.

Thank you

Answer 1

If the samples are iid given the parameters, e.g. $x_i$ are coin tosses and $\theta=p=P(\text{Head})$, then $p(x_2|x_1,\theta)=p(x_2|\theta)$ and we'll use this fact below: $$\begin{align}p(\theta|x_1,x_2)&=\frac{p(x_2|\theta,x_1)p(\theta|x_1)p(x_1)}{\int p(x_2|\theta,x_1)p(\theta|x_1)p(x_1) d\theta}=\frac{p(x_2|\theta,x_1)p(\theta|x_1)}{\int p(x_2|\theta,x_1)p(\theta|x_1)d\theta}\\&=\frac{p(x_2|\theta)p(\theta|x_1)}{\int p(x_2|\theta)p(\theta|x_1)d\theta}\end{align}$$