An urn contain 5 balls, $ \theta $ white and $ 5 - \theta $ green. The experiment consists in grab 2 balls from the urn and register the pair $(x_1, x_2)$, where $x_i = 1$ if we observe a white ball and $x_i = 0$ otherwise. What is the bayes estimator $ \theta^* $ for $ \theta$ considering the squared loss function? (i.e $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 $ \theta $ and $ \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