(maximum-a-posteriori-estimation)=
# Maximum a-posteriori estimation

Instead of maximizing $p(f\mid u)$ we want to maximize, the posterior
```{math}
p(U=u\mid F=f)
```
The probability of the unknown image being $u$ under observation of the
noisy image being $f$.

````{prf:theorem} Bayes' Theorem
Let $(\Omega,E,p)$ be a probability space and let $A,B \in E$ be two
events with $p(B)>0$. Then,
```{math}
p(A|B) = \frac{p(B|A)\cdot p(A)}{p(B)}.
```

````

````{prf:proof} 
One can show this theorem by extending the fracture with $1$ and simply
inserting the definition:
```{math}
p(A|B) = \frac{p(A\cap B)}{p(B)} = \frac{p(A\cap B) \cdot p(A)}{p(B)\cdot p(A)}= \frac{p(B|A)\cdot p(A)}{p(B)}.
```
 ◻

````

With Bayes' theorem we can calculate the *maximum a-posteriori
estimation* (MAP-estimation) using again the negative log
```{math}
\begin{aligned}
    \operatorname{arg\,min}_{u\in X}-\log{p(u|f)} &= \operatorname{arg\,min}_{u\in X}-\log{p(f|u)}-\log{p(u)} + \log{p(f)}\\
    &= \operatorname{arg\,min}_{u\in X}-\log{p(f|u)}-\log{p(u)}\,.
\end{aligned}
```

The MAP-estimation gives us an unknown image, which fits the
obeservation $f$ with maximal probability. To calculate it with Bayes'
theorem, we only need the probability of $u$ under observation of $f$
and the probability of the event of $u$, which is called the *prior*. We
introduce the *Gibbs prior* $p(U=u) = e^{-\alpha R(u)}$ with a positive
weighting parameter $\alpha>0$ and a function $R(u)$, which introduces
prior information of the problem. We calculate now the MAP-estimator
with Gibbs prior

```{math}
\begin{aligned}
\operatorname{arg\,min}_{u\in X}-\log{p(u|f)}= \operatorname{arg\,min}_{u\in X}{ -\log{p(f|u)} + \alpha R(u)},
\end{aligned}
```

Revisiting the calculated conditional probabilities for the different
noise models {prf:ref}`ex:noise_models`, we have the following
MAP-estimators for the Gaussian and Poisson noise models.
```{math}
\begin{aligned}
\operatorname{MAP}_G (f) &= \operatorname{arg\,min}_{u\in X}\frac{\|f-u\|_2^2}{2\sigma^2} + \alpha R(u) \\
\operatorname{MAP}_P (f) &= \operatorname{arg\,min}_{u\in X}\langle \mathbf{1}, u\rangle - \langle f, \log(u)\rangle + \alpha R(u) 
\end{aligned}
```
We can generalize these MAP-estimators with Gibbs priors to the
following class of optimization problems
```{math}
\operatorname{arg\,min}_{u\in X}{\underset{\color{green}\text{data fidelity term}}{D(u_i, f)} + \underset{\color{orange}\text{regularization term}}{\alpha R(u)}}
```

It remains for us to study, based on a given data fidelity term and
regularizer,

-   the existence and uniqueness of minimizers $u \in X$,

-   the regularity of the solution $u$,

-   the calculation or approximation of $u$.

-   Which regularizer $R$ and data term $D$ is the best model for our
    application?

We will answer these questions in the upcoming chapters by applying
tools from variational calculus and convex analysis.