05 Optimisation Problem

The idea is that we define a loss function and then try to find the optimum of that.

Example

• Find the estimate of true temperature
• Measurements:

$$\begin{array}{r}{x}_1=19,x_2=17,x_3=24\\ \mathbf{x}=(x_1,x_2,x_3)\end{array}$$

• Here loss is difference of one temperature w.r.t to other
• Strategy: find estimate that minimises loss

$$\begin{array}{r}{x}^{\ast }=\arg \underset{\hat{x}}{\min }L(\hat{x},\mathbf{x})\\ L(\hat{x},\mathbf{x})=\sum _{i=1}^n|x_i-\hat{x}|\end{array}$$