How to change priors in Bayesian estimation, if we get to know previous work was wrong? Assuming we are doing a task to find how runs will be scored some match and for that, we have assumed some prior, now what if we find out midway through the process on which we built our priors was inaccurate due to various factors. How should we the strategize for the further process and at the same time not dump the previous work done so far though the initial priors were not correct?
 A: Suppose you are trying to predict the outcome of an election on Proposition A, which needs 60% Yes votes to pass. Let $\theta$ be the unknown proportion of the electorate in favor.
Maybe you have a favorable view of the probability of success, so you
choose the prior $\mathsf{Beta}(7,3),$ which implies $P(\theta > 0.6) = 0.77$
1 - pbeta(.6, 7, 3)
[1] 0.768213

You discover that early results from a public opinion poll are running only
about 50-50, and you begin to have doubts about your prior distribution.
There are a couple of possibilities (among many):

*

*Your prior is grossly wrong.

*Your prior is OK and the early returns from the poll are not yet reliable.

In either case, you should just wait for the final poll results.
If they
show 735 in favor out of 1000, then the likelihood function
is proportional to $\theta^{735}(1-\theta)^{265},$ the beta posterior
distribution is $\mathsf{Beta}(742,268),$ a 95% Bayesian interval
estimate for $\theta$ is $(0.71, 0.76),$ your prior doesn't look so
bad in retrospect, and
the proposition looks on target for success.
qbeta(c(.025,.975), 742, 268)
[1] 0.7070060 0.7614208

Alternatively, the final poll results may show only 520 out of 100 in favor,
then the posterior distribution is $\mathsf{Beta}(527,483),$ a
95% Bayesian interval estimate for $\theta$ is $(0.49, 0.55),$ the data have
overwhelmed your less-that-ideal prior, and it seems proponents of Proposition A
have some work to do before election day.
qbeta(c(.025,.975), 527, 483)
[1] 0.4909563 0.5525263

Of course, it is possible, if ordinarily imprudent, to begin with such a strong
prior that only a huge amount of data could overwhelm it. Also, if the people determining
the prior distribution are truly clueless about Proposition A and its prospects,
then they might admit their indecision and give a relatively noninformative prior
such as $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1).$ In that case the prior
distribution will have almost no effect on the eventual prediction interval.
Finally, if it seems for some reason that the first probability interval estimate is not adequate, then the posterior distribution might become the prior distribution for another round of
Bayesian estimation based on a subsequent poll.
