# Estimating $\sigma_i$ according to maximum likelihood method

Let be a Bayesian multivariate normal distribution classifier with distinct covariance matrices for each class and isotropic, i.e. with equal values over the entire diagonal and zero otherwise, $$\mathbf{\Sigma}_i=\sigma_i^2\mathbf{I},~\forall i$$.

How can I compute the equation for estimating the parameter $$\sigma_{i}$$ by the maximum likelihood method? Here $$\sigma_{i,j}$$ is is the covariance between $$x_i$$ and $$x_j$$. So $$\sigma_i$$ is just the variance of $$x_i$$.

Attempt:

Suppose $$\mathcal{X}_i = \{x^t_i\}^N_{t=1}$$ i.i.d, $$x_i^t$$ is in the class $$C_i$$ and $$x_i^t \sim \mathcal{N}(\mu, \sigma^2)$$.

Do I have to find the log-likelihood under $$p(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[ -\frac{(x-\mu)^2}{2 \sigma^2}\right]$$, find the derivative and put it equal to $$0$$ to find the maximum?

EDIT

Suppose my data points are $$m$$-dimensional, and I have $$K$$ classes.

• Comments are not for extended discussion; this conversation has been moved to chat. – nbro Sep 25 '20 at 20:40
• The problem here is how will you estimate the MLE. It is pretty difficult estimation problem (computationally) and thus has tradeoffs in space vs computational complexity. proceedings.mlr.press/v75/hazan18a/hazan18a.pdf A good place to start your problwm. Although I would warn, its a pretty difficult problem requiring a very good theoretical knowledge. – user9947 Sep 25 '20 at 20:55
• How can your data points be $m$ dimensional if they follow a univariate normal distribution? – David Ireland Sep 25 '20 at 21:52