SECTION S1 CONVEXITY OF THE LOG LIKELIHOOD FUNCTION IN

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Section S1: Convexity of the log likelihood function

Section S1: Convexity of the log likelihood function

In this section we show that log likelihood of the observed symptom profile counts is a convex function of the incidence parameters (the matrix of its second derivatives is negative definite) if the distribution of symptoms matrix has full rank (equaling the number of possible pathogens). This in particular implies that the EM iterations will converge to the unique maximum likelihood estimate (given the matrix ) if the limit is in the interior.

Let be the population proportions of symptomatic infections associated with the different pathogens on some week (including the asymptomatic proportion ) and let be the symptom profile counts on that week (including no symptoms ). We extend the matrix to be an matrix having a bottom row and a rightmost column of zeroes except for the bottom right element, which is 1 – the new matrix, which we still call , is the distribution of possible outcomes given different states, and it still has full rank.

The multinomial log likelihood (minus the constant term of the logarithm of the multinomial coefficient) is

The matrix of its second derivatives with respect to the variables is

SECTION S1 CONVEXITY OF THE LOG LIKELIHOOD FUNCTION IN

One readily sees that , where is the matrix with

SECTION S1 CONVEXITY OF THE LOG LIKELIHOOD FUNCTION IN .

This means that for any vector , the second derivative of in the direction of at point (along a line ) is

and the latter is strictly negative if by the assumption that has full rank (because the columns of , which are the rows of , are proportional to the columns of , hence has a zero kernel).

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Tags: convexity of, function, section, convexity, likelihood