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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?

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    $\begingroup$ This sounds like homework, so maybe tell us what have you tried and where are you stuck? $\endgroup$
    – Tim
    Jun 10 at 22:10
  • $\begingroup$ Thank you for the reply! I've just edited the question. $\endgroup$
    – Empty
    Jun 11 at 20:42
  • $\begingroup$ 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. $\endgroup$
    – jbowman
    Jun 12 at 1:26

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