**3.1 One parameter case**

When *θ* is a scalar quantity, then, we can also do the following computations:

• Compute the value of *θMed* such that:

$$P(\theta > \theta\_{Med}) = P(\theta < \theta\_{Med}) \tag{13}$$

which is called the *median value*. Its computation needs integration:

$$\int\_{-\infty}^{\theta\_{\rm{Mad}}} p(\theta|\mathbf{g}) \, \mathbf{d}\theta = \int\_{\theta\_{\rm{Mad}}}^{\infty} p(\theta|\mathbf{g}) \, \mathbf{d}\theta \tag{14}$$

• Compute the value *θα*, called *α* quantile, for which

$$P(\theta > \theta\_a) = \mathbf{1} - P(\theta < \theta\_a) = \int\_{\hat{\theta}\_a}^{\infty} p(\theta | \mathbf{g}) \, \mathbf{d}\theta = \mathbf{1} - a \tag{15}$$

• Region of high probabilities: [needs integration methods]

$$\left[\hat{\theta}\_1, \hat{\theta}\_2\right] : \int\_{\hat{\theta}\_1}^{\hat{\theta}\_2} p(\theta | \mathbf{g}) \, \mathrm{d}\theta = 1 - a$$

Bayes rule and Bayesian estimation can be illustrated as follows:

$$p(\theta) \rightarrow \begin{array}{c} p(g|\theta) \\ \downarrow \\\\ \text{Bayes} \\\\ \hline \\\\ \hline \end{array} \rightarrow p(\theta|g) \rightarrow \begin{cases} \widehat{\theta}\_{\text{MAP}} \\ \widehat{\theta}\_{\text{PM}} \\ \widehat{\theta}\_{\text{Mcd}} \end{cases}$$

Two main points are of great importance:


This last problem becomes more serious with multi parameter case.
