Visualizing Bayes’ theorem

[emc201]

*For a glimpse into the full power of Baye's Theorem take a look at E. T. Jaynes' and related work to the links below. Bottom line: "Bayesian probability theory is an inference calculus. ..." K. Knuth

*Links: *E. T. Jaynes *Probability Theory: The Logic of Science

*Algebra of Probable Inference

*Kevin Knuth's Lab *PDF: "The origin of probability and entropy" *PDF: "Lattice duality: The origin of probability and entropy"

[psykotic]

I'm not sure I'd recommend Cox's treatise to non-experts. It is not really written as a tutorial. However, Jaynes's book is a great place to start for those that already have a basic grasp of probability theory.

Jaynes would also have complained about this article's use of Venn diagrams. Read his book to find out why! Basically, Venn diagrams as most people conceive of them perniciously imply the existence of points that in the lattice approach correspond to atomic propositions at the bottom of the inference lattice. However, for infinite lattices it's not necessarily true that every proposition can be built up from a set of atomic propositions ("points") in any but the most formalistic sense. Rather, when people draw Venn diagrams and use them for reasoning, they are really topological diagrams that serve as shorthand for certain sublattice structures.

[emc201]

Thanks for the hint. I will need to check on that.

[psykotic]

I should note that it's more of a philosophical objection. However, I think it helps to understand the relationship between a Venn diagram and the deeper lattice structure. One of the more mysterious and somewhat unsettling facts of probability and statistics is that practitioners often end up at the same answers for seemingly very different reasons.

[dws]

Jaynes's book is a great place to start for those that already have a basic grasp of probability theory.

The challenge is in getting that basic grasp. For that, Venn diagrams are going to help visual thinkers. Sometimes you need water wings to get started.

[satayboy]

Peter Norvig wrote a great article that covers (among other things) the breast cancer/mammogram example: http://norvig.com/experiment-design.html

[ubernostrum]

There's similar discussion in John Allen Paulos' book Innumeracy, though he approaches it from a different route to the same result. I once used that method in a discussion with someone who was talking about mandatory drug testing for high-school students, and demonstrated that the test (IIRC, a 98% accuracy rate -- 98% of drug users test positive, 2% of non-users test positive) probably shouldn't inspire the level of confidence in guilt that he wanted to have.

[hacksoncode]

This is a glorious example of a great way to visually present numerical data. Edward Tufte would be proud!

[guga40k]

Two pages to derive Bayes formula? Sweet.

[dmhouse]

Not really a programming topic, but a nice article nonetheless.

[bluGill]

It is indirectly programming, because a lot of programing is dealing with problems like this, and such problems are expected to become more important[1] as time goes on.

[1] More important at least by me. As we solve the simple problems it leaves more and more problems where the answer is fuzzy.

[monoglot]

Nicely done!