Bayes rule in practice: definition and parameter estimation

A slecture by ECE student Chuohao Tang

Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.

Content:

1 Bayes rule for Gaussian data

2 Procedure

3 Parameter estimation

Given a data set $ \mathbf{X}=(\mathbf{x}_1,...,\mathbf{x}_N)^T $ in which the observations $ \{{\mathbf{x}_n}\} $ are assumed to be drawn independently from a multivariate Gaussian distribution (D dimension), we can estimate the parameters of the distribution by maximum likelihood. The log likelihood function is given by

By simple rearrangement, we see that the likelihood function depends on the data set only through the two quantities

These are the sufficient statistics for the Gaussian distribution. The derivative of the log likelihood with respect to $ \mathbf{\mu} $ is

and setting this derivative to zero, we obtain the solution for the maximum likelihood estimate of the mean

Use similar method by setting the derivative of the log likelihood with respect to $ \mathbf{\Sigma} $ to zero, we obtain and setting this derivative to zero, we obtain the solution for the maximum likelihood estimate of the mean

4 Example

5 Conclusion

## Questions and comments

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