Extension onto multi-variate distributions

The mathematics generalises to multi-variate distributions easily. It merely requires a re-interpretation of $\theta$, x and R in equation () to their multi-variate counterparts. The problem of finding the optimal region R can be re-interpreted as finding the boundary points of the optimal region. With a 2-dimensional parameter-space, the boundary to the optimal region is a one-dimensional curve. In general, for a n-dimensional parameter-space, the boundary points will form a (n-1)-surface.

However, it has been difficult to find an algorithm to determine the optimal region or its boundary. This problem falls under the class of problems known as non-linear programming. Furthermore, general non-linear equation solvers (such as Newton-Raphson) cannot be used unmodified as they generally assume the existence of only one solution. While a numerical solver for MMLD has not been implemented in higher dimensions, there are two strategies to tackle this problem.

A generalised non-linear equation solver can be used, provided there only exists a single solution. Therefore, an additional constraint that that limits the solution to a single point on the boundary of the region can be used. Then by varying the constraint, the boundary can be re-sampled until a certain level of detail is reached. For example, the boundary could be intersected with a straight line that stretches from an interior point to an exterior point of the region. A limitation of this example is that the line must not intersect the boundary more than once. If the region can be guaranteed to be concave, then this limitation does not apply.

Alternatively, there is a mathematical method called the ``continuation method'' [#!Chu:private!#] that relies on the continuity of the boundary surface. This method starts off with a single boundary point and finds nearby boundary points. Continuation methods often involve the use of partial derivatives, which is acceptable as our likelihood function is analytical within the region.

Non-linear programming techniques are either (computationally) inefficient, incomplete or both. Hence it is extremely beneficial to determine whether any further assumptions (eg concavity of the region) can be made about the region or the likelihood function. If further assumptions about the problem can be justified, more efficient or complete methods would be available. Some useful proofs would show that -

1.

The optimal region R (and non-optimal R_i) are simply-connected.

2.

The optimal region R (and non-optimal R_i) are concave.

3.

The negative-log-likelihood function (over the domain) has only a single point-of-inflection (maxima, minima, saddle-points) within the domain of the parameter.

These three properties are currently unknown, but it would be extremely useful in finding a more efficient and appropriate algorithm. The proof of 2 or 3 would imply the proof of 1. The proof of any of these three properties would enhance the usefulness of MMLD in higher dimensions.