06 Maximum Likelihood

• Our measurements are noisy and follow a normal distribution:

$$p(x_i;\mu ,\sigma )=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(\mu -x_i{)}^2}{2{\sigma }^2})$$

• $\mu$ is true temperature, and $\sigma$ is amount of noise
• Measurements are conditionally independent:

$$p(\mathbf{x},\mu ,\sigma )=\prod _{i=1}^{n}p(x_i;\mu ,\sigma )$$

The Normal/Gaussian Distribution

• Many things are approx. normal distributed, e.g., height.
• Go-to-distribution for many modelling application.