The Search for a Great Bayesian Example

When we teach about the Pythagorean Theorem, we almost always, at some point, use a 3-4-5 triangle. The numbers are friendly, and they work. We don’t usually make this explicit, but I bet that many of us also carry that triangle around in our own heads as an internal prototype for how right triangles work—and we hope our students will, too. (The sine-cosine-1 triangle is another such prototype that develops later.)

In teaching about (frequentist) hypothesis testing, I use the Aunt Belinda problem as a prototype for testing a proportion (against 0.5). It’s specific to me—not as universal as 3-4-5.

Part of this Bayesian quest, I realized, is to find a great example or two that really make Bayesian inference clear: some context and calculation that we can return to to disconfuse ourselves when we need it.

The Paper Cup Example

Here’s the one I was thinking about. I’ll describe it here; later I’ll explain what I think is wrong with it.

I like including empirical probability alongside the theoretical. Suppose you toss a paper cup ten times, and 8 of those times it lands on its side. At that point, from an empirical perspective, P( side ) = 8/10. It’s the best information we have about the cup. Now we toss it again and it lands on its side. Now the empirical probability changes to 9/11.

How can we use a Bayesian mechanism, with 8/10 as the prior, to generate the posterior probability of 9/11?

It seemed to me (wrongly) that this was a natural. Continue reading The Search for a Great Bayesian Example

Early Bump in the Bayesian Road: a Search for Intuition

Last time, I introduced a quest—it’s time I learned more about Bayesian inference—and admitted how hard some of it is. I wrote,

The minute I take it out of context, or even very far from the ability to look at the picture, I get amazingly flummoxed by the abstraction. I mean,

P(A \mid B) = \frac{P(A)P(B \mid A)}{P(B)}

just doesn’t roll of the tongue. I have to look it up in a way that I never have to with Pythagoras, or the quadratic formula, or rules of logs (except for changing bases, which feels exactly like this), or equations in kinematics.

Which prompted this comment from gasstationwithoutpumps:

I find it easiest just to keep coming back to the definition of conditional probability P(A|B) = P(A & B) / P(B). There is no complexity here…(and more)

Which is true, of course. But for this post I’d like to focus on the intuition, not the math. That is, I’m a mathy-sciencey person learning something new, trying to record myself in the act of learning it. And here’s this bump in the road: What’s up with my having so much trouble with a pretty simple formula? (And what can I learn about what my own students are going through?) Continue reading Early Bump in the Bayesian Road: a Search for Intuition