# Compute the Maximum A Posteriori (MAP) estimate of θ

How can I compute the Maximum A Posteriori (MAP) estimate of $$\theta$$ with those informations: a discrete random variable y with values in {1, 2, . . . , N} has a Binomial distribution depending on the unknown probability $$\theta \in (0,1)$$ of the form $$p(y=k|θ)=\binom{N}{k}θ^{k}(1-θ)^{N-k}$$. I've to compute the Maximum A Posteriori (MAP) estimate of $$\theta$$ based on a single observation y assuming a prior density on $$\theta$$ to be Beta distribution $$B(x; a, b)=\frac{1}{B}x^{a-1}(1-x)^{b-1}$$ where $$x \in (0, 1)$$, $$a, b > 0$$ and $$B$$ a normalization parameter. I also know that the mode of the Beta distribution is $$\hat{x}=\frac{a-1}{a+b-2}$$.
My idea is to calculate $$\arg \max_{θ}[p(y=k|θ)p(θ)]$$ but I don't know if it is the right way, and I don't even know how to proceed.
Thanks!
EDIT: I tried to compute the MAP estimate of $$\theta$$ using the theoretical definition: $$\hat{θ}_{MAP}=\arg \max_{θ}[p(y=k|θ)p(θ)]$$.
So I obtained: $$\hat{θ}_{MAP}=\arg \max_{θ}[\binom{N}{k}θ^{k}(1-θ)^{N-k}\frac{1}{B}θ^{a-1}(1-θ)^{b-1}]$$. At this point I don't know how to proceed. Is there any analytical method to find the value of θ that maximize this function? Can the mode of the Beta distribution be an useful information?

• This sounds like homework, so maybe tell us what have you tried and where are you stuck?
– Tim
Jun 10 at 22:10
• Thank you for the reply! I've just edited the question. Jun 11 at 20:42
• Try simplifying your last equation as follows: a) throw out everything that isn't a function of $\theta$, b) combine the terms involving $\theta$ appropriately. Then the resolution of your problem should be clear. Jun 12 at 1:26