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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 :

MLE method

but with Bayesian estimation on the same problem, (It says, with p(theta) = theta(1-theta))

Bayesian method

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?

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  • "Bayesian estimation" is a somewhat imprecise term. There is "maximum a-posteriori estimation" and "posterior mean estimation". Your book seems to use the former. Does your textbook give a precise statement of what Bayesian estimation is? – kimchi lover Apr 16 '23 at 13:39

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