Bayesian statistics example

Question

I am learning Bayesian statistics for a data mining module. We were given a set of practice questions but no answers and I want to make sure that I am going about Bayes in the right way.

Suppose Manchester United were about to play Arsenal in the Premiership and you assess the probability of Manchester United winning to be 0.7. You also feel that if they did win, there is a probability of 0.9 that your local pub will be packed with fans celebrating their club’s victory. Alternatively, if they loose, you believe that there is still a probability of 0.6 that the pub will be packed with fans albeit for drowning their sorrows. As someone not interested in football, you enjoy a quiet day at work and on the way home, notice a big crowd in the pub. Can you assume that Manchster United won the game? If so, what is the probability of them having won the game?

Attempted solution:

let $p(w)$ denote prob of manchester united winning let $p(f)$ denote prob of pub being full

$p(w|f)=\frac{p(f|w)*p(w)}{p(f)}$ , where $p(f)=p(f|w)*p(w)+p(f|¬w)*p(¬w)$

$p(f)= 0.9*0.7+0.6*0.3=0.81$

$\rightarrow$$p(w|f)=\frac{0.9*0.7}{0.81}=0.78$

So prob of Manchester untied having won is 0.78.

I don't think we can assume they won just because there were people in the pub as there was a 0.6 prob of them drowning their sorrows.

Have I got this right or have I misunderstood Bayesian statistics?

Thanks very much

Answer 1

You have correctly applied Bayes rule but this is a far cry from understanding Bayesian statistics. Bayesian statistics is a way of doing statistical inference that is different the frequentist approach. Suppose you want to get an interval estimate of the mean of a normal distribution. You first put a prior distribution on the mean then you collect data. Bayes' theorem is used to relate the prior distribution for the mean to the posterior distribution by posterior is proportional to prior times likelihood. The posterior distribution is then used for inference. An interval estimate for the mean that is the Bayesian analog of a 95% confidence interval is a 95% credible interval. You get this by picking points equal distant in each direction from the mean of the posterior distribution and far enough so that the integral from the lower endpoint to the upper endpoint is 0.95. This is what Bayesian statistics is about. It is much more than just Bayes theorem. Bayes theorem is an uncontroversial mathematical result in probability. Bayesian statistics is a paradigm for statistical inference and is somewhat controversial because it assigns probability distributions to unknown parameters.

Answer 2

Both your reasoning and your computations are correct.