## A Bayesian Example: Two coins, three heads.

As laid out (apparently not too effectively) here, I’m on a quest, not only finally to learn about Bayesian inference, but also to assess how teachable it is. Of course I knew the basic basics, but anything in stats is notoriously easy to get wrong, and hard to teach well. So you can think of this in two complementary ways:

• I’m trying to ground my understanding and explanations in basic principles rather than leaping to higher-falutin’ solutions, however elegant; and
• I’m watching my own wrestling with the issues, seeing where I might go off-track. You can think of this as trying to develop pedagogical content knowledge through introspection. Though that sounds pretty high-falutin’.

To that end, having looked critically at some examples of Bayesian inference from the first chapters of textbooks, I’m looking for a prototypical example I might use if I were teaching this stuff.  I liked the M&Ms example in the previous post, but here is one that’s simpler—yet one which we can still extend.

There are two coins. One is fair. The other is two-headed. You pick one at random and flip it. Of course, it comes up heads. What’s the probability that you picked the fair coin?

## 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

## Modeling Digression: Coin Weights

Too much philosophy, let’s get some data!

Part of the motivation for the recent posts on modeling is that I’m writing a paper for a friend. It takes off from the hexnut weight data and cubic model that appear in The (still unpublished, sigh) Model Shop. More on that soon, because I’ve found some interesting features in hexnut data. But first, another friend sent me a link to this post on doghousediaries I now share with you. Check it out.

Because I had just been doing hexnuts, I immediately thought about coins as an alternative: how do you suppose the weight of coins (Physics people: I will be reporting weight in grams. We both know that’s mass, not weight. We can handle that, right?) is related to their size?

I mean, hexnuts look more or less geometrically similar (they’re not, but that’s the other story), so you’d think their mass is cubic in the linear size: a nut that fits a half-inch bolt ought to be eight times the weight of one that fits a quarter-inch bolt. And that model fits pretty well.

So as teachers, we look for situations that stretch that understanding. EGADs has you cut out cardboard squares or circles and weigh them; since the difference in size is two-dimensional, the weight-size relationship is quadratic.

But what about coins? They look kind of two-dimensional, but the big ones tend to be thicker. So are they cubic or quadratic or something in between? Well. Whenever I come back from some other country, I usually have leftover coinage, which gets saved in that jar in the top drawer. And I have a good scale and (I know, I’m a measurement geek) a decent micrometer. So it’s an empirical question.