Parameter Estimation for Maximum Likelihood (Univariate Gaussian)

A univariate Gaussian is a special case of a multivariate Gaussian where the data is one-dimensional. The parameter vector is $\theta =[\mu ,{\sigma }^2{]}^T$.

The ML estimation process is analogous to the multivariate case:

• The ML estimate of the mean ($\hat{\mu }$) is the sample mean (average) of the scalar training data points.

  $$\hat{\mu }=\frac{1}{T}\sum _{\tau =1}^T{o}_{\tau }$$

• The ML estimate of the variance ($\widehat{{\sigma }^2}$) is the sample variance of the training data.

  $$\hat{\Sigma }=\frac{1}{T-1}\sum _{\tau =1}^T(o_{\tau }-\hat{\mu })(o_{\tau }-\hat{\mu }{)}^T$$

  Note: Using a denominator of T-1 provides an unbiased estimate.