Beta Distribution Problem

Question

So I have the made up situation where I have a coin of bias $0.4$ so the coin is slightly more likely to land on tails. Now, I have data where after $50$ flips, I get $18$ heads. So if I have the belief that the coin is fair then after $50$ flips, I should get $25$ heads and $25$ tails. So my prior would be

Beta$(25,25)$

And that would make my posterior

Beta$(25+18,25+32)$ $=$ Beta$(43,57)$

That's all good and I have plots in python for this. Now I want to say about if you choose a really bad prior like Beta$(40,10)$ which would make the posterior Beta$(58,42)$. I plot the graph and I get something off like I would expect.

Now the main question I want to ask is how can I then improve this? If I have a very poor prior like this and I plot the posterior, what can I do to improve the posterior? If I try just adding a scale factor like Beta$(58+20,42+20)$ kind of improves it but it doesn't bring the posterior close to the $\theta = 0.4$ line.

Answer 1

Bayesian inference is based on taking both the prior distribution and the data seriously. Each has its appropriate effect on the posterior distribution. If you somehow know that the success probability is $\theta =0.4,$ then how do you arrive at the prior distribution BETA(50, 50)?

Do you have knowledge of previous tosses of the coin that showed half heads and half tails. If you just have a vague feeling that the coin 'looks OK' after quick inspection, then maybe the prior should be something like BETA(4, 4). If you just arrived from Mars and don't even know what a 'coin' is, then maybe you would want a non-informative prior like BETA(1,1) = UNIF(0,1) or BETA(.5, .5).

Posterior distributions BETA(43. 57) and BETA(58, 42) give the following 95% Bayesian probability intervals, respectively, using R. Notice that both intervals contain success probability 0.5.

qbeta(c(.025,.975), 43, 57)
[1] 0.3349894 0.5276638
qbeta(c(.025,.975), 58, 42)
[1] 0.4824441 0.6745238

Data from 1000 tosses of the coin would overwhelm either the priors you propose; that is, the prior would not have much influence on the posterior. Finally, I have no idea what you mean by 'scale factor'.