# In Bayesian statistics does posterior probability become prior probability when new observation is made?

First you have prior probability say 50% . When observation is made we use that to update our prior probability = posterior probability 66%. If another observation is made our prior probability 66% which we would update with new observation right? and we continue this process?

I just want to clarify if my understanding it correct.

• Yes. Iterative Bayesian updating is exactly that. – usεr11852 May 22 '20 at 0:18

Just one thing, prior is a probability distribution, not a value like 50% (prior may be also improper.)

As an example with the Binomial and Beta prior:

Starting with a prior distribution $$Be(1,1)$$ (this is the uniform distribution).

We see 5 outcomes, where 3 are positive and 2 negative.

The posterior is now $$Be(1+3,1+2)$$ = $$Be(4,3)$$. This is the prior for the next time.

We see another 13 outcomes, 8 positive, 5 negative.

The posterior is now $$Be(4+8,3+5)$$ = $$Be(12,8)$$.

We see 125 outcomes, 80 positive, 45 negative.

The posterior is now $$Be(12+80,8+45)$$=$$Be(92,53)$$.

Here is each curve. 