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The formula for calculating the MAP estimate of a particular parameter, $p$, is given by the following: $p^{MAP} =$ argmax $P(p)P(p|x)$.

Now I am trying to do a question where I am told the prior distribution $P(p)$ and am given that the prior distribution is a Beta distribution. Using conjugate priors, I can then determine that the posterior $P(p|x)$ should be an "updated" Beta distribution according to the number of successes and failures.

At this point, I can take the product of $P(p)P(p|x)$ and go ahead and use calculus to find the value of $p$ that corresponds to the argmax.

However, the question hints that I should solve this question by maximising the log posterior with respect to $p$. I do not understand this suggestion since if I only maximise $P(p|x)$ doesn't it ignore the information about $p$ contained in $P(p)$?

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With $p$ as the parameter and $x$ as the data, the MAP solution maximises the posterior $\mathrm{prob}(p | x, \mathcal{I})$:

\begin{align} p_\mathrm{MAP} &= \underset{p}{\operatorname{argmax}} \mathrm{prob}(p | x, \mathcal{I}) \\ &= \underset{p}{\operatorname{argmax}} \frac{\mathrm{prob}(x | p, \mathcal{I}) \: \mathrm{prob}(p | \mathcal{I})}{\mathrm{prob}(x | \mathcal{I})} \\ &= \underset{p}{\operatorname{argmax}} \mathrm{prob}(x | p, \mathcal{I}) \: \mathrm{prob}(p | \mathcal{I}) \\ &= \underset{p}{\operatorname{argmax}} \ln\left[\mathrm{prob}(x | p, \mathcal{I}) \: \mathrm{prob}(p | \mathcal{I})\right] \end{align}

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  • $\begingroup$ what is $\mathcal{I}$? $\endgroup$ – user11128 Feb 16 '19 at 12:05
  • $\begingroup$ @user11128 It looks like just a shorthand for all the other information other than $p$ and $x$ that would go into calculating these probabilities. The key thing to get from this answer, which I think could be improved by highlighting this or saying it plainly in some way, is that the MAP solution maximizes $\Pr(p|x)$, which we in practice do by maximizing $\Pr(p)\Pr(x|p)$ (the prior of $p$ times the likelihood of $x$ given $p$), which is proportional to the posterior $\Pr(p|x)$, and truly in practice maximize the log of that (it increases accuracy and sometimes computational efficiency). $\endgroup$ – duckmayr Feb 16 '19 at 12:24
  • $\begingroup$ @user11128 $\mathcal{I}$ is the background information you use to assign the probabilities. $\endgroup$ – CarbonFlambe--Reinstate Monica Feb 16 '19 at 12:45

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