# Bayesian formulation of disease test: What likelihood function to use? [closed]

Disease Test: In this classic example of Bayes' theorem for discrete events, one would like to determine the posterior probability of the patient having the disease, given that they test positive, which turns out to be a function of prevalence, sensitivity and specificity.

It is commonplace to extend Bayes' theorem to functions for the prior, likelihood, posterior and indeed the evidence, not least in order to enable propagation of uncertainties on the input parameters, and then to use sampling e.g. MCMC, nested sampling etc. to solve for the full posterior.

Q. What form of the likelihood function should one adopt? What would be candidates for the prior distribution?

Thanks!

## closed as unclear what you're asking by Xi'an, kjetil b halvorsen, Michael Chernick, Stephan Kolassa, mdeweyFeb 5 '18 at 10:15

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## 1 Answer

You could use a multinomial likelihood with a Dirichlet prior. Your posterior will be a Dirichlet distribution and the solution is closed form so you do not need MCMC.

• Excellent - thanks for the pointer - seems to work! The question has been "put on hold" but I am not sure how to make it any clearer (you understood it for example....) - any ideas? – jtlz2 Feb 6 '18 at 8:25
• @jtlz2 this question does not depend upon a Bayesian formulation. In fact, the Frequentist formulation is far simpler and easier to understand. The Bayesian version could readily be argued is an unnecessary formalism UNLESS you had certain concerns such as a legislature investing money in a process. In that case, Bayesian methods are coherent and Frequentist are not. The downside to using the Bayesian method is that you lose the guarantee against false positives greater than $1-\alpha$ and the unbiasedness. The other reason to use the Bayesian method is if you have real prior knowledge. – Dave Harris Feb 6 '18 at 21:09
• @jtlz2 to be honest, I had to take a second or third look at the problem to resolve it in my own mind. – Dave Harris Feb 6 '18 at 21:10