I am currently reading Jaynes' Probability Theory, The Logic of Science and am still trying to absorb everything. On page 190, he poses the following intriguing question, paraphrased here.
Suppose you fell asleep on the train and upon waking up, you arrive at an unknown station.
All you can see is a taxicab with the number 27. What is then your guess at the number N of taxis in the town (assuming they are numbered from 1 consecutively).
The "straightforward" answer would be 27. I came up with my own solution, see below, and I wonder whether it is a correct way of thinking about the problem. Also, since I am lacking formal training, I am interested in a more systematic analysis. In particular I'd like to have someone sketch the "true Bayesian" solution to this, with all the quantities explicitly stated and not glossed over. Thank you.
My thinking is as follows. Suppose the probability of encountering any taxi is 1/N. Then the probability of seeing a taxicab with number at most x is x/N. So, seeing taxicab 27, the number N is at least 27 with probability 1. Continuing, 27 is with 90% probability from the 0-0.9 quantile, from which I think I can conclude that N>=30 with 90% probability. Likewise, I'd say that N>=54 with 50% probability, etc.
This is known in the literature as the German tank problem. The discussion at Wikipedia is long and complex, but at the risk of greatly oversimplifying things, if you only see one taxi, numbered $m$, then the minimum variance unbiased estimator is $2m-1$.