The difference between a likelihood slice and a profile
Exercise
What is the difference between a likelihood slice and a profile? What is the consequence of this difference for the statistical interpretation of these plots? How should you decide whether to compute a profile or a slice?
Solution
A likelihood surface with two parameters can be visualized as a mountain, with the elevation of the mountain at each spatial location corresponding to the value of the likelihood at this coordinate pair of parameters.
Suppose we look at the mountain from the south, at a large distance.
The profile likelihood is the silhouette of the mountain.
The slice of the likelihood through the maximum likelihood estimate is the elevation at an east-west cross-section through the peak of the mountain.
The profile and the slice could be very different. For example, imagine the situation if the mountain’s peak has sheer cliffs to the east and west, but a gradual ridge descending south-east to north-west.
For a mountain with elliptical contours with axes running north-south and east-west, the slice and profile are the same. This corresponds to the likelihood function for independent Gaussian measurements on two parameters.
The profile likelihood has useful statistical properties arising from its relationship to likelihood ratio tests and Wilks’ theorem. It can therefore be used to construct confidence intervals.
A slice cannot usually be used to build confidence intervals.
A slice is much quicker to compute than a profile, since it involves likelihood evaluation along a range of parameter values whereas the profile involves likelihood maximization along this range.
A slice can be useful as a relatively quick, informal investigation of the likelihood. Profiles may be calculated in a later stage of the analysis, when we are ready to apply all the computational resources at our disposal.
If only one parameter is being estimated, a slice and a profile are the same thing!
