In the urn model, it does not make sense to talk about the probability of p being greater than a certain value because p is a fixed value.
With Bayesian statistics, we assume that p is in fact random, which allows us to calculate probabilities related to p .
Hierarchical models describe variability at different levels and incorporate all these levels into a model for estimating p .
Bayes' Theorem
Bayes' Theorem states that the probability of event A happening given event B is equal to the probability of both A and B divided by the probability of event B:
\[Pr(A∣B)=\frac{Pr(B∣A)Pr(A)}{Pr(B)}\]
Bayes' Theorem shows that a test for a very rare disease will have a high percentage of false positives even if the accuracy of the test is high.
Cystic fibrosis test probabilities
The Hierarchical Model
Hierarchical models use multiple levels of variability to model results. They are hierarchical because values in the lower levels of the model are computed using values from higher levels of the model.
The posterior distribution allows us to compute the probability distribution of p given that we have observed data Y .
By the continuous version of Bayes' rule, the expected value of the posterior distribution p given Y=y is a weighted average between the prior mean μ and the observed data Y : \[E(p∣y)=B\mu+(1−B)Y\] where \(B=\frac{\sigma^2}{\sigma^2+\tau^2}\)
The standard error of the posterior distribution \[SE(p∣Y) = \sqrt{\frac{1}{1/\sigma^2+1/\tau^2}}\] .
This Bayesian approach is also known as shrinking. When σ is large, B is close to 1 and our prediction of p shrinks towards the mean \(\mu\). When σ is small, B is close to 0 and our prediction of p is more weighted towards the observed data Y .