Parameter identification: A new perspective
Judea Pearl · eScholarship (California Digital Library) · 2011
TECHNICAL REPORT R-276 January 2001 Parameter Identification: A New Perspective (Second Draft) Judea Pearl Cognitive Systems Laboratory Computer Science Department University of California, Los Angeles, C A 90024 [email protected] Introduction and Preliminary Terminology: A model M is a set of structural equations w i t h (zero or more) free parameters, p,q,r,..., that is, unknown parameters whose values are to be estimated from a combination of as- sumptions and data. The assumptions embedded i n such a model are of several kinds: (1) zero (or fixed) coefficients i n some equations, (2) equality or inequality constraints among some of the parameters and (3) zero covariance relations among error terms (also called disturbances). Some of these assumptions are encoded implicitly i n the equations (e.g., the absence of certain variables in an equation), while others are specified explicitly, using expressions such as: p = q or cov{ei, ej) = 0. A n instantiation of a model M is an assignment of values to the model's parameters; such instantiations w i l l be denoted as mi, m^ etc. The value of parameter p in instantiation m i of M w i l l be denoted as p{m\). Every instantiation of model M gives rise to a unique covariance matrix a {mi), where a is the population covariance matrix of the observed variables. Definition 1 (Parameter identification) A parameter p in model M is identified if for any two instantiations have: p{mi) = p{m ) whenever a {mi) = o (m ) Definition 2 (Model identification) A model M is identified iff all parameters of M are identified of M,mi and m^, we