Section S1 Convexity Of The Log Likelihood Function In

section s1: convexity of the log likelihood function in this section we show that log likelihood of the observed symptom profile counts is

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

One readily sees that , where is the matrix
with
.
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).