Why no binomial distribution in the Euro Problem

[heneryville]

In the Euro problem, when calculating the likelihood of the entire set at once, it seems like this should use the binomial distribution. The binomial distribution calculates what the odds are of seeing K instances in N draws if the probability is P, and it seems like that's exactly what the likelihood should be, with N being tails + heads, K being heads, and P being x.

How does this likelihood function differ from a binomial?

[AllenDowney]

Good question. I have a TODO on exactly this point: https://github.com/AllenDowney/ThinkBayes/blob/master/book/book.tex#L2497 I will address this in the next edition. Thank you!

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On Sun, Jan 28, 2018 at 10:02 PM, Mitchell Harris ***@***.***> wrote: In the Euro problem, when calculating the likelihood of the entire set at once <https://github.com/AllenDowney/ThinkBayes/blob/master/book/book.tex#L2462-L2466>, it seems like this should use the binomial distribution. The binomial distribution calculates what the odds are of seeing K instances in N draws if the probability is P, and it seems like that's exactly what the likelihood should be, with N being tails + heads, K being heads, and P being x. How does this likelihood function differ from a binomial? — You are receiving this because you are subscribed to this thread. Reply to this email directly, view it on GitHub <#14>, or mute the thread <https://github.com/notifications/unsubscribe-auth/ABy37Q_q9URy7qKqJz3Rb_Wa-NIOujOqks5tPTTBgaJpZM4RwASu> .

[yongduek]

I also had the same question, and concluded that:

1. Binomial distribution must be used in calculating the likelihood for 250 trials.
2. The likelihood for each hypothesis $x$ (1 out of 101) has the same binomial term: 250C140
3. The binomial terms disappear when normalization for posterior distribution is applied.
4. So, binomial term does not have to be included as long as the posterior is concerned.

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So... my question is ... where is the explanation about the evidence postponed to the next chapter?

Compared to the explanation in MacKay's book, the approach in this book is much more clearer and simpler for me to understand. Many thanks again.

[ricardoV94]

@yongduek I was going through the same reasoning as you and a post on StackExchange led me to the formal explanation of this 'puzzle'. It follows from the likelihood principle that a bayesian inference about the parameter p will be the same regardless of which likelihood is used (bernoulli, binomial or negative_binomial) in this case. This argument is used in MacKay's book when he criticizes how frequentist inference changes when you assume different likelihood functions for the same data.