
Bayes estimator
An urn contain 5 balls, $\displaystyle \theta $ white and $\displaystyle 5  \theta $ green. The experiment consists in grab 2 balls from the urn and register the pair $\displaystyle (x_1, x_2) $ , where $\displaystyle x_i = 1 $ we observe a white ball and $\displaystyle x_i = 0$ otherwise. What is the bayes estimator $\displaystyle \theta^* $ for $\displaystyle \theta $ considering the squared loss function? (i.e $\displaystyle l(\theta,\theta^*) = (\theta  \theta^*)^2 $ )
I can't figure out which posterior distribution I should use or even if I need to use one. I calculated my loss function considering all the possible values for $\displaystyle \theta $ and $$\displaystyle \theta^* $ but I can't calculate the risk function without the posterior function.
Can someone help me with it?
I can't find out what to do with the ordered pair, I just calculated the probability of each pair

Re: Bayes estimator
Hey Giiovanna.
Under squared loss, the best estimator is always the mean (its a standard result).
Hint: Take a look at the conjugate prior for a hypergeometric distribution and use this to form your prior. Once you have your prior*likelihood distribution you can use this to get your posterior and then your mean from this distribution.