I just worked out question 3.18 myself, and found an answer of 9.31. Which differs by one decimal position from the answer in this repo.
I found this other solution manual here that like me also arrived a 9.31:
https://github.com/ArthurZC23/Machine-Learning-A-Probabilistic-Perspective-Solutions/blob/master/3/18.pdf
I think the mistake here is in the step where the uniform pmf gets plugged in, resulting in a P(D | M_1) = 1/(N+1).
Assuming that M_1 here refers to the alternative hypothesis H1, this doesn't seem right.
This seems like it should be:
P(D | H_1) = \int_{0}^{1} L(\theta | D) P(\theta | H_1)
With L(\theta | D) the likelihood function and P(\theta | H_1) the uniform prior on \theta as specified in the alternative hypothesis.
Calculating that integral, we find a denominator of 110 rather than 11.
In the numerator, strictly speaking this integral exists there too:
P(D | H_0) = \int_{0}^{1} L(\theta | D) P(\theta | H_0)
However, in the numerator that doesn't matter because P(\theta | H_0) has a Dirac point mass on 0.5, and thus the integral drops off and we are left with only the Bernoulli likelihood function.
I just worked out question 3.18 myself, and found an answer of 9.31. Which differs by one decimal position from the answer in this repo.
I found this other solution manual here that like me also arrived a 9.31:
https://github.com/ArthurZC23/Machine-Learning-A-Probabilistic-Perspective-Solutions/blob/master/3/18.pdf
I think the mistake here is in the step where the uniform pmf gets plugged in, resulting in a P(D | M_1) = 1/(N+1).
Assuming that M_1 here refers to the alternative hypothesis H1, this doesn't seem right.
This seems like it should be:
P(D | H_1) = \int_{0}^{1} L(\theta | D) P(\theta | H_1)
With L(\theta | D) the likelihood function and P(\theta | H_1) the uniform prior on \theta as specified in the alternative hypothesis.
Calculating that integral, we find a denominator of 110 rather than 11.
In the numerator, strictly speaking this integral exists there too:
P(D | H_0) = \int_{0}^{1} L(\theta | D) P(\theta | H_0)
However, in the numerator that doesn't matter because P(\theta | H_0) has a Dirac point mass on 0.5, and thus the integral drops off and we are left with only the Bernoulli likelihood function.