This is based upon my solution of Exercise 2.3, page 18, R. Christensen, W. Johnson, A. Branscum, T. E. Hanson, Bayesian Ideas and Data Analysis, Chapman & Hall, 2011. The purpose is to show how information latent in a set of data can be revealed by Bayesian inference or, as it is sometimes called in this case, Bayesian inversion.

There’s data about numbers of cars manufactured on different days, and data regarding numbers of those cars which are defective.

It would be interesting to know what’s the proportion overall that are defective. Well, it’s

Then, it’s interesting to ask, what’s the probability of a defective car given that it’s Friday, or

Applying Bayes Rule, this can be calculated as

The same can be asked about Tuesday, or

Now, this data is known. I buy a car. It’s defective. What are the odds it was manufactured on a Friday as opposed to a Tuesday?

My point? This needed be limited to problems regarding defective cars. Rather than calculating p-values from mysteriously posed t-tests applied to statistical populations for which the assumptions regarding t-tests are questionable, such as might arise in determining whether a temperature trend is statistically important, with a bit more sophistication, the odds of there being a trend versus their not being a trend can be calculated. Moreover, different models can be compared in similar ways.

Professor John Kruschke explains in part why and how:

Vehtari and Ojanen say a lot more, and Congdon illustrates the technique in detail.