I'm working through Think Bayes (free here: http://www.greenteapress.com/thinkbayes/) and I'm on exercise 3.1. Here's a summary of the problem:
"A railroad numbers its locomotives in order 1..N. One day you see a locomotive with the number 60. Estimate how many locomotives the railroad has."
This solution is found with the likelihood function and exponential prior like so:
class Train(Suite): def __init__(self, hypos, alpha=1.0): # Create an exponential prior Pmf.__init__(self) for hypo in hypos: self.Set(hypo, hypo**(-alpha)) self.Normalize() def Likelihood(self, data, hypo): if hypo < data: return 0 else: return (1.0/hypo)
Conceptually this is saying, if we see a train number larger than one of our hypotheses (1...1000) then every hypothesis that's smaller has a zero chance of being correct. The rest of the hypotheses have a 1/number_of_trains chance of showing us a train with this number.
In the exercise I'm working on the author then adds on a little extra. This assumes there's only one company. In real life however you'd have a mixture of big and small companies and bigger companies (both equally likely). However, this would mean that you're more likely to see a train from a bigger company since they'd have more trains.
Now the question is how to reflect this in the likelihood function?
This isn't Stack Overflow so I'm not really asking for coding help, but instead perhaps just help about how I might think about this problem in terms of a likelihood function.