Maximum Likelihood Degree and PI
From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 07/15/04
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Date: Thu, 15 Jul 2004 00:21:04 +0000 (UTC)
From Osher Doctorow mdoctorow@comcast.net
F. Catanese, S. Hosten, A. Khetan, and B. Sturmfels, in "The maximum
likelihood degree," arXiv:math.AG/0406533 v1 25 Jun 2004, discuss
some of the latest developments in Algebraic Statistics, which
intersects quite a bit with my recent threads here.
The "AG" in the arXiv listing refers to Algebraic Geometry, which
readers should look up in the Front for the Mathematics ArXiv as
keywords under Algebraic Geometry, and goes back to the early 1990s
where some of the pioneering papers appeared. It is also interesting
as an exercise to see who came up with original ideas and who made
"variations on a theme," although some people who do variations on
a theme eventually learn to develop original ideas later on under
certain circumstances.
Three papers cited by Catanese et al are L. Garcia, M. Stillman, and
B. Sturmfelds, "Algebraic geometry of Bayesian networks," math.AG/
0301255, J. Symbolic Comput., to appear; L. Pachter and B. Sturmfels,
"Tropical geometry of statistical models, q-bio.QM/0311009, Proc.
Natl. Acad. Sci. USA, to appear; and G. Pistone, E. Riccomagno, and
H. P. Qynn, "Algebraic statistics: computational commutative algebra
in statistics," Chapman and Hall, Boca Raton, Florida, 2001. How-
ever, some of the older references cited have involved very Creative
people who have remained at the top of the field, and readers should
look at the paper themselves. Of course, where there are Bayesian
networks, a PI interpretation cannot be far behind (see above).
Readers will find themselves introduced to or looking up concepts
like sheaves, degrees of algebraic functions, generating functions,
independence models, irreducible projective varieites, logarithmic
singularities, normal crossing divisors, Chern class, projective
d-space, smooth toric varieties, Newton polytopes, bounded regions,
arrangements of hyperplanes, linear forms, resolution of singularit-
ies, etc. The hyperplanes {f_i = 0} are reminiscent of PI express-
ions, and the authors derive a formula for the Maximum Likelihood
(ML) degree when the f_i's are Laurent polynomials, as for example:
1) f_i = ai + bi(theta_1) + gi(theta_2) + di(theta_1)(theta_2)
for i = 1, 2, 3, 4, where i is a subscript and _ also represents a
subscript and the polynomials are in variables theta_1 and theta_2
with coefficients bi, gi, di. The authors hope that algorithms
can eventually be developed to better solve the critical equations
dlog(f) = 0 for f = f_1^u1 f_2^u2 ... f_n^un and so on.
The authors point out that their paper was motivated by recent
papers on ML degree in statistics and computational biology, includ-
ing a paper on phylogenetic models in 2003, a paper on unidirected
graphical models in 2002, a paper on ML degree of a Gaussian graph-
ical model (2004), a paper on ML degree of mixture models (2004),
where specific references to these are listed in their reference
section. For all these papers, the exact numerical values of the
ML degree were found.
So what is the ML degree? It is the degree of the algebraic
function that is an optimal solution theta-hat to a problem which
is an algebraic function of the data u, namely the problem of
maximizing f_1(theta)^u1 f_(theta)^u2...f_n(theta)^un for theta in
an appropriate subset of R^d. The paper begins with discrete
data and later does some work on certain continuous data.
Osher Doctorow
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