I'm confused with an example in MLE and Bayesian estimation.
It's about 'coin tossing', where we toss a coin n times, and the probability of observing head is theta. 'theta' is an unknown parameter in this case.
The textbook says, we denote x as :
1 (when head), 0 (when tail)
and D to be the sample set of all xs defined above. then with MLE :
but with Bayesian estimation on the same problem, (It says, with p(theta) = theta(1-theta))
How is the solution in the Bayesian method possible (theta_hat is a fixed value)? What I learnt was, the estimated parameter theta_hat would be a fixed value in MLE estimation, but in Bayesian estimation, the estimated parameter would be a random variable and is not a fixed value. Also, I don't get that the estimated parameter is derived by solving the gradient equation = 0. How is this possible?