I love Bayes -- and you can too!

No truism is nearly so elegant as, or responsible for more deep insights than, Bayes's Theorem.

I've linked to a couple of teaching tools that I use in my evidence course. One is a Bayesian calculator, which Kw Bilz at UIUC first came up with & which I've tinkered with over time.

The second is a graphic rendering of a particular Bayesian problem. I adapted it from an article by Spiegelhalter et al. in Science. 

In my view, the "prior odds x likelihood ratio = posterior odds" rendering of Bayes is definitely the most intuitive and tractable. It's really hard to figure out what people who use other renderings are trying to do besides frustrate their audience or make them feel dumb, at least if they are communicating with those who aren't used to manipulating abstract mathematical formuale.  As the graphic illustrates, the "odds" or "likelihood ratio" formalization, in addition to being simple, is the one that best fits with the heuristic of converting the elements of Bayes into natural frequencies, which is an empirically proven method for teaching anyone -- from elementary school children (or at least law students!) to national security intelligence analysts-- how to handle conditional probability.

If you don't get Bayes, it's not your fault.  It's the fault of whoever was using it to communicate an idea to you.

References

Sedlmeier, P. & Gigerenzer, G. Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General 130, 380-400 (2001).

Spiegelhalter, D., Pearson, M. & Short, I. Visualizing Uncertainty About the Future. Science 333, 1393-1400 (2011).
 

Wheaton, K.J., Lee, J. & Deshmukh, H. Teaching Bayesian Statistics To Intelligence Analysts: Lessons Learned. J. Strategic Sec. 2, 39-58 (2009).

Comments

[Chuanpeng Hu]

thanks for your links, Prof.

[Jon Baron]

Another device is here:

http://www.sas.upenn.edu/~baron/900/bayes.xm.

This is not self-explanatory. It is for classroom demos by a teacher who explains it, and for students who have heard and read the explanation.

And to my knowledge it works properly only in Firefox. You can change the numbers and see how they affect the different cells and the posteriors.

[Dan Kahan]

@Jon Baron: cool! Thanks! Notwithstanding what I said about "likelihood ratio" rendering in post, I recognize that in fact there is surprising amount of heterogeneity in comprehension styles when it comes to graphic (& nongraphic) presentation of data. Probably the lesson of the empirical work here is -- come w/ a well-stocked heuristic toolkit & keep fishing around until you find the one that fits the nut. Pretty sure that's what Spiegelhalter would say!

@Chuanpeng Hu-- you are most welcome!

[NiV]

Take logarithms and call it 'information'.

The information post-experiment is the information you have prior to the experiment plus the information you get from the experiment (the log likelihood ratio).

Information theory gives us some of the deepest insights in physics. It's a little more distant from probabilities, so you could argue about just how intuitive it is, but if you squint your eyes...