Independence in exponential fields

Zilber constructed a class of exponential�fields CFSK,CCP whose models have exponential-algebraic properties similar to the classical complex field with exponentiation Cexp. In this thesis we study this class and the more general classes ECFSK, also defined by Zilber, and ECF, studied by Zilber and Kirby. We investigate stable-like behaviour modulo arithmetic in these classes by developing a unique independence relation for each class, and in ECF we use this relation to examine types. We provide an exposition of exponential fields that is more model theoretic and type-oriented than preceding work. We then investigate the types in ECF that are orthogonal to the kernel. New ideas presented include a characterisation of these types, and the definition of a grounding set; these results allow us to�find su�fficient conditions to prove that a type over a set uniquely extends to a type over the smallest strong ELA-sub�field containing that set. For each class we define a ternary relation on subsets, and prove that these relations are independence relations, with properties akin to non-forking independence in first order theories. Applying work of Kangas, Hyttinen and Kes�al�a, we prove that in ECFSK our independence notion is the unique independence relation for this class, and that our independence notion in ECFSK,CCP is exactly the canonical independence relation for this class derived from the pre-geometry. Assuming the conjecture known as CIT, we use our independence relation in ECF to prove that types orthogonal to the kernel are exactly the generically stable types

Independence relations for exponential fields

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP$_1$-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.Comment: 25 page

On the decomposition of Generalized Additive Independence models

The GAI (Generalized Additive Independence) model proposed by Fishburn is a generalization of the additive utility model, which need not satisfy mutual preferential independence. Its great generality makes however its application and study difficult. We consider a significant subclass of GAI models, namely the discrete 2-additive GAI models, and provide for this class a decomposition into nonnegative monotone terms. This decomposition allows a reduction from exponential to quadratic complexity in any optimization problem involving discrete 2-additive models, making them usable in practice

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if both of the sign patterns $(1,-1)$ and $(-1,1)$ are represented in the sample for every pair of variables; in particular, this implies that the MLE may exist with $n=d$ observations, in stark contrast to unrestricted binary exponential families where $2^d$ observations are required. Finally, we provide a novel and globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders