On-line estimation with the multivariate Gaussian distribution

We consider on-line density estimation with the multivariate Gaussian distribution. In each of a sequence of trials, the learner must posit a mean mu and covariance Sigma; the learner then receives an instance x and incurs loss equal to the negative log-likelihood of x under the Gaussian density parameterized by (mu, Sigma). We prove bounds on the regret for the follow-the-leader strategy, which amounts to choosing the sample mean and covariance of the previously seen data.