Estimation of Probability Measures Using Aggregate Population Data 177
as the choice of the nodes {θ
k
}
M
k=1
is an open research problem. We remark
that values of M under 50 have bee n s tudied computationally in [4, 11].
It is clear from the statement of Theorems 5.4.4 and 5.4.5 that the unique-
ness of the minimizer
ˆ
P
N∗
is not guaranteed (as there could be multiple
conve rgent subsequences). In particular, in the implementation of the ap-
proximations on P
M
(Ω
θ
), for a fixed set of node s {θ
k
}
M
k=1
, it is possible to
establish first-order neces sary conditions for the uniqueness of the computa-
tional problem [4, p. 102]. But these should not be confused with conditions
for global uniqueness of a minimizing measure P
0
. Unlike the non-parametric
maximum