Teaching Bayesianism

JQuinton

I've had a bit of success with getting people to understand Bayesianism at parties and such, and I'm posting this thought experiment that I came up with to see if it can be improved or if an entirely different thought experiment would be grasped more intuitively in that context:

Say there is a jar that is filled with dice. There are two types of dice in the jar: One is an 8-sided die with the numbers 1 - 8 and the other is a trick die that has a 3 on all faces. The jar has an even distribution between the 8-sided die and the trick die. If a friend of yours grabbed a die from the jar at random and rolled it and told you that the number that landed was a 3, is it more likely that the person grabbed the 8-sided die or the trick die?

I originally came up with this idea to explain falsifiability which is why I didn't go with say the example in the better article on Bayesianism (i.e. any other number besides a 3 rolled refutes the possibility that the trick die was picked) and having a hypothesis that explains too much contradictory data, so eventually I increase the sides that the die has (like a hypothetical 50-sided die), the different types of die in the jar (100-sided, 6-sided, trick die), and different distributions of die in the jar (90% of the die are 200-sided but a 3 is rolled, etc.). Again, I've been discussing this at parties where alcohol is flowing and cognition is impaired yet people understand it, so I figure if it works there then it can be understood intuitively by many people.

Say there is a jar that is filled with dice. There are two types of dice in the jar: One is an 8-sided die with the numbers 1 - 8 and the other is a trick die that has a 3 on all faces. The jar has an even distribution between the 8-sided die and the trick die. If a friend of yours grabbed a die from the jar at random and rolled it and told you that the number that landed was a 3, is it more likely that the person grabbed the 8-sided die or the trick die?

It's very straightforward in frequentist interpretation: half the people pick normal die, one in 8 rolls 3, so 1/16 of original people roll 3 off normal die, while 1/2 roll 3 off trick die, for total of 9/16 rolling 3. 1/16 with normal die in 9/16 that roll 3, here's your probability. Trivial stuff people should be able to reinvent if they skip or forget. 4th or 5th grade math at most. Too bad there's no good training at the early enough age.

Train people to think straight and the Bayes will pop up; train people to do Bayes and they'll think wrong with Bayes.

edit: actually, what's up with this local trope of "Bayesianism" as opposed to "Frequentism"? The math abstracts out the philosophical detail of whenever probability is a degree of belief or product of convergence of long term trials.

actually, what's up with this local trope of "Bayesianism" as opposed to "Frequentism"? The math abstracts out the philosophical detail of whenever probability is a degree of belief or product of convergence of long term trials.

Where do you get the idea that it's a local trope? Knowledgable and well-respected people in the field consider these foundational issues important, e.g., Brad Efron and Andrew Gelman.

You can make an argument that the philosophical details wash out as long as you're operating on a fully specified probability s... (read more)

7pragmatist14y

There are practically relevant considerations that emerge from the philosophical distinction between Bayesianism and frequentism. If you have an epistemic conception of probability, then it makes sense to talk about the probability distribution of a theoretical parameter, such as the mean of some variable in a population. If you're a frequentist, though, this usually does not make sense. The variable itself has a relative frequency associated with each of its values, but it makes no sense to talk of the relative frequency of the mean of the variable. So on a frequentist conception of probability, you wouldn't assign a probability distribution over the theoretical parameter. The parameter is not treated as a random variable. The upshot is that frequentists focus on likelihood functions, which a Bayesian would describe as giving the probability of observed data conditional on the theoretical parameter. Bayesians, on the other hand, look for posterior probability distributions, the distribution of the theoretical parameter conditional on observed data. Bayesians report the epistemic probability that the parameter value is in a certain interval, given the data. For frequentists, the interval is the random variable, and the unknown parameter is treated as fixed. So the probabilities they report are the relative frequency (in an ensemble of experiments) with which the confidence interval constructed using observed data will contain the theoretical parameter. So the philosophical difference has important methodological consequences in statistics. These are explored at length in many advanced textbooks in theoretical statistics, such as this one. The distinction isn't just a local trope; it's an important foundational distinction in statistics.

2David_Gerard14y

Eliezer got this straight from Jaynes.