Radioactive Cookies, Hierarchical Parameter Estimation, and The Beta Distribution, Continued

Today I Learned:
1) Here's a nice little thought experiment to understand the properties of different kinds of dangerous radiation: You have three cookies. One emits alpha radiation, one emits beta radiation, and one emits gamma radiation. You have to put one in your shirt pocket, hold one in your hand, and eat one. Which do you put where?

The best answer: Hold the alpha cookie in your hand. Your skin will block alpha particles, but if they get in your body they're really bad. Put the beta cookie in your pocket -- it's a little stronger, but your clothes should block most of it. The gamma cookie, you eat -- it's going to hurt you, but nothing short of a sheet of lead is going to protect you at all anyway, so it doesn't matter where you put it.

2) Hierarchical parameter estimation is really cool, but it consistently requires more parameters than you might expect.... (Hierarchical parameter estimation, by the way, is a kind of parameter estimation where the parameter of interest takes on different values in different experiments.

My favorite example is in preference estimation, like when a company wants to know the most tastiest amount of salt to put in a jar of salsa is (call it O_salt, for Optimum Salt). You might imagine the company performing a bunch of experiments on people to figure out the true O_salt, and average the results of the experiments to find O_salt. The trouble is, different people have different preferences, which are distributed in some way around O_salt. The better model is that each person P has their own optimum salt concentration P_salt, where P_salt is distributed around some O_salt. )

3) ...a bit more about the Beta distribution. The Beta distribution is a probability distribution over the range [0,1], and it's often used to model probabilities of probabilities -- for example, the probability of a (real world) coin landing heads on a flip might be beta-distributed tightly around 0.5. The cool thing about the beta distribution is that it has two parameters, alpha and beta (sorry about the term overload!), which can be the number of successes and failures observed for a Bernoulli trial, respectively, giving the "correct" probability distribution over the success of that Bernoulli trial (for some sense of "correct" which I haven't bothered to learn).

For example, take the coin flip example. If you flip a coin 100 times and it comes up heads 44 times and tails 56 times, then the probability distribution over the possible probabilities of the coin coming up heads is beta-distributed with alpha = 44 and beta = 56.

If that didn't make sense, I direct you to the place I learned this, which has a more thorough explanation: http://ift.tt/1P1UGYy