**3.2 Multi-parameter case**

If we have more than one parameter, then *θ* ¼ ½ � *θ*1, ⋯, *θ<sup>n</sup>* <sup>0</sup> *:* The Bayes rule still holds:

$$p(\theta|\mathbf{g}) = \frac{p(\mathbf{g}|\theta)p(\theta|\mathbf{g})}{p(\mathbf{g})} \tag{16}$$

Now, again, we can compute:

• The Expected A Posteriori (EAP):

$$
\hat{\boldsymbol{\theta}}\_{\rm PM} = \int \boldsymbol{\theta} p(\boldsymbol{\theta} | \mathbf{g}) \, \mathrm{d}\boldsymbol{\theta}, \tag{17}
$$

but this needs efficient integration methods.

• The Maximum A Posteriori (MAP):

$$\hat{\theta}\_{MAP} = \underset{\theta}{\text{arg max}} \; \{ p(\theta | \mathbf{g}) \} \tag{18}$$

but this needs efficient optimization methods.

• Sampling and exploring [Monte Carlo methods]

$$
\boldsymbol{\theta} \sim p\left(\boldsymbol{\theta} \| \boldsymbol{\mathsf{g}}\right),
$$

but this needs efficient sampling methods.

• We may also try to localize the region of the highest probability:

$$P(\theta \in \Theta) = \int\_{\Theta} p(\theta | \mathbf{g}) \, \mathrm{d}\theta = 1 - a \tag{19}$$

for a given small *α*, but this problem may not have a unique solution.
