Aaron Bergman comments on Aaron Bergman’s Quick takes

[Aaron Bergman]

The recent 80k podcast on the contingency of abolition got me wondering what, if anything, the fact of slavery’s abolition says about the ex ante probability of abolition—or more generally, what one observation of a binary random variable $X$ says about $p$ as in

Turns out there is an answer (!), and it’s found starting in paragraph 3 of subsection 1 of section 3 of the Binomial distribution Wikipedia page:

A closed form Bayes estimator for p also exists when using the Beta distribution as a conjugate prior distribution. When using a general $\text{Beta}(\alpha ,\beta )$ ⁡ as a prior, the posterior mean estimator is...

${\hat{p}}_b=\frac{x+\alpha }{n+\alpha +\beta }$

[...]

For the special case of using the standard uniform distribution as a non-informative prior, $\text{Beta}(\alpha ,\beta )=U(0,1)$ , the posterior mean estimator becomes: ${\hat{p}}_b=\frac{x+1}{n+2}$

Don’t worry, I had no idea what $\text{Beta}(\alpha ,\beta )$ was until 20 minutes ago. In the Shortform spirit, I’m gonna skip any actual explanation and just link Wikipedia and paste this image (I added the uniform distribution dotted line because why would they leave that out?)

So...

Cool, so for the $n=1$ case, we get that if you have a prior over the ex ante probability space$[0,1]$ described by one of those curves in the image, you...

• 0) Start from ‘zero empirical information guesstimate’ $\text{E}\left[\text{Beta}\right(\alpha ,\beta \left)\right]=\frac{\alpha }{\alpha +\beta }$

• 1a) observe that the thing happens $(x=1)$, moving you, Ideal Bayesian Agent, to updated probability ${\hat{p}}_b=\frac{1+\alpha }{1+\alpha +\beta }>\frac{\alpha }{\alpha +\beta }$ OR

• 1b) observe that the thing doesn’t happen $(x=0)$, moving you to updated probability ${\hat{p}}_b=\frac{\alpha }{1+\alpha +\beta }<\frac{\alpha }{\alpha +\beta }$

In the uniform case (which actually seems kind of reasonable for abolition), you...

• 0) Start from prior $E\left[p\right]=1/2$

• 1a) observe that the thing happens, moving you to updated probability ${\hat{p}}_b=2/3$

• 1a) observe that the thing doesn’t happen, moving you to updated probability ${\hat{p}}_b=\frac{1}{3}$

[Peter Wildeford]

The uniform prior case just generalizes to Laplace’s Law of Succession, right?

[Aaron Bergman]

In terms of result, yeah it does, but I sorta half-intentionally left that out because I don’t actually think LLS is true as it seems to often be stated.

Why the strikethrough: after writing the shortform, I get that e.g., “if we know nothing more about them” and “in the absence of additional information” mean “conditional on a uniform prior,” but I didn’t get that before. And Wikipedia’s explanation of the rule,

Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observed one success and one failure for sure before we even started the experiments.

seems both unconvincing as stated and, if assumed to be true, doesn’t depend on that crucial assumption

[Robi Rahman]

The last line contains a typo, right?

[Aaron Bergman]

Fixed, thanks!