# Maximum A Posteriori Estimate

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 $$P(p)$$?

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}

• what is $\mathcal{I}$? – user11128 Feb 16 '19 at 12:05
• @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). – duckmayr Feb 16 '19 at 12:24
• @user11128 $\mathcal{I}$ is the background information you use to assign the probabilities. – CarbonFlambe--Reinstate Monica Feb 16 '19 at 12:45