Probabilistic Modelling and Reasoning Solutions for Tutorial 7 Spring 2018

Solution. Since the logarithm is strictly monotonically increasing, the maximiser of the log-likelihood equals the maximiser of the likelihood. It is easier to take derivatives for the log-likelihood function than for the likelihood function so that the maximum likelihood estimate is typically determined using the log-likelihood. Given the algebraic expression of `(θ), it is simpler to work with the variance v = σ2 rather than the standard deviation. (In the lecture notes, we used the variable η to denote the transformed parameters. We could have written η = σ2, but v is a more natural notation for the variance.) Since σ > 0 the function v = g(σ) = σ2 is invertible, and the invariance of the MLE to re-parametrisation guarantees that σ̂ = √ v̂.