Identifiability In Statistics

Identifiability is a crucial property in statistics, as it is a primary requirement for models in order to be correctly estimated.

A statistical model is a collection of probability distributions. A model can be defined by a function from a set of parameters: in this case it is called parametric model, and the function itself is called parametrization map. Identifiability is a ‘’good property” for the parametrization map: a model is identifiable if this function is one to one, that is, each probability distribution in the model is given only by one parameter. If identifiability fails, it means that there is not a unique way to locate a specific probability in the model. When there are multiple choice of parameters for every single distribution

However, there are other, more subtle, possible cases: in fact other weak form of identifiability have been introduced in literature such as local and generic identifiability. The objective of this work is to detect such non-trivial cases and explore the consequences of the

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lack of identifiability for models with latent variables.

1.2 Outline of this work

We begin this work by giving a brief historical overview on the problem of the identifiability in the statistical literature.

In Chapter 2 we introduce the latent variable models, and we will give a few motivating examples in order to show their usefulness. Latent variable models make use of one or more non-observable random variables to explain the variability in the data. The archetipical model is the latent class model, where the observable variables are assumed independent when conditioned to the latent variable. Is is therefore necessary to adopt a parametrization that allows to easily define conditional independencies. This role is taken by graphical log-linear models, of which we will give an essential overview in the rest of the

A few examples are then given, in order to show the consequences of these conditions.

Chapter 4 presents the first original results of the thesis: first, we present the characterization given by Stanghellini and Vantaggi (2013, [26]) for the identifiability of graphical models. Then, we move further with a characterization of the identifiability for a different class of models: hierarchical models with interactions of order at most 2. This result is complete: we have found a simple necessary and sufficient condition for models with full rank matrices, based on the topology of the graphs encoding all the independences. It turned out that 5 observed variables are sufficent for achieving local identifiability in this class of models. Finally, in Chapter 5, we present some results on hierachical models with interactions of higher order. We give a general recursive procedure to assess if a log-linear hierarchical model is generically identifiable. Some simulation results suggest that generic identifiability holds when the number of observed variables is twice the maximal order of the interactions included in the model.

Most of the original reasearch in this thesis has been done together with

Barbara Vantaggi, whom I heartily