Statisticians Find They Can Agree: It’s Time to Stop Misusing P-Values

Suppose our prior distribution of beliefs is 0.5 that the probability is exactly 1/2, versus 0.5 that the probability of a boy is some value P which is equally likely to be any value from 0 to 1.

In the first case, the probability of 6 boys and 1 girl is 0.5^7 = 1/2^7. In the second case the probability of 6 boys and 1 girl is P^6(1-P) = P^6 - P^7. The integral from 0 to 1 of P^6 - P^7 is 1/6-1/7 = 1/42. Each case also has a priori odds of 1/2 of holding true.

After observing 6 boys and 1 girl, the first case now has probability (0.5/2^7)/(0.5/2^7 + 0.5/42) = 1/(1 + 64/21) = 21/129 = 0.162790697674419. The second case now has probability 1 - this, which is 0.837209302325581. Furthermore if the second case is true, P is no longer uniformly distributed. In fact its density is now proportional to P^6-P^7.

So the posterior distribution is now going to be:

With probability 21/129, exactly 0.5. And otherwise any value P from 0 to 1 with a probability density of 108/129*(P^6-P^7)/42.

Given this prior and this set of observations, any other answer is wrong. Given a different prior you would get a different posterior, but as long as the prior gives a constant probability of male/female, the only fact that matters is how many boys and girls there are.

The order of births can only start to matter if you start with a prior that gives different probabilities of different genders based on prior events. Even then it is hard to come up with a realistic scenario in which the plans of the parents would make an order of magnitude difference in the posterior distribution.

What would you do if you were studying a process that had not been previously studied, so you had no previous experience/data on which to base your a prior? If you wait, and choose a prior based on the data you observe then you are using an empirical Bayes method [1].

[0]: https://en.wikipedia.org/wiki/Prior_probability#Uninformativ...

[1]: https://en.wikipedia.org/wiki/Empirical_Bayes_method

It's also common to have a complex, multi-level model setup which has a few hyperparameters (say, gamma distribution shapes) which are set basically arbitrarily. The idea being that the posteriors for the lowest levels (closest to data) will be learned/fitted, but the hyper parameters at the top-level of the model are fixed. This is very common in spatial statistics.

It's also common to make lots of (conditional) independence assumptions, just because they are convenient. Such as diagonal covariances, or conditional independence between "separate" elements of a model. But these independence assumptions are often just convenient or based on intuition, but not re-checked. Or, if they are checked and found wanting, it's "left for future work".

These practices are defensible, and attackable. But choosing convenient priors out of a bag of standard priors, without reference to the problem at hand, is very common.

You can't actually say unless you either (1) roll the die more times, or (2) assume something about the probability that I gave you an all-7's die to begin with.

Doing (2) is useless, because that exactly the question we are trying to answer.

For example, suppose I perform this experiment all the time and I know that I give an all-7's die only 1% of the time. With this new information, you could actually calculate the probability of an all-7's die given a 7 roll. Of all the possible outcomes, you could add up the ones where I gave you an all-7s die and the ones where I gave you a normal die but you just rolled 7. Then you could divide that by the total number of possible outcomes.

But this would give you a totally different number than if I give you an all-7's die 99% of the time. And the problem is that you don't have any information about what kind of die you have before you roll it. You're trying to figure out which world we live in -- one where your hypothesis is true or one where it's not.

(I am pretty sure that what I wrote above is true. But one thing I'm not as clear on is how multiple rolls of the die actually can establish confidence percentages. How many rolls does it take to actually establish confidence? Would love to hear from any stats experts about that.)