8 research outputs found
Exact Dimensionality Selection for Bayesian PCA
We present a Bayesian model selection approach to estimate the intrinsic
dimensionality of a high-dimensional dataset. To this end, we introduce a novel
formulation of the probabilisitic principal component analysis model based on a
normal-gamma prior distribution. In this context, we exhibit a closed-form
expression of the marginal likelihood which allows to infer an optimal number
of components. We also propose a heuristic based on the expected shape of the
marginal likelihood curve in order to choose the hyperparameters. In
non-asymptotic frameworks, we show on simulated data that this exact
dimensionality selection approach is competitive with both Bayesian and
frequentist state-of-the-art methods
The Dependence of Routine Bayesian Model Selection Methods on Irrelevant Alternatives
Bayesian methods - either based on Bayes Factors or BIC - are now widely used
for model selection. One property that might reasonably be demanded of any
model selection method is that if a model is preferred to a model
, when these two models are expressed as members of one model class
, this preference is preserved when they are embedded in a
different class . However, we illustrate in this paper that with
the usual implementation of these common Bayesian procedures this property does
not hold true even approximately. We therefore contend that to use these
methods it is first necessary for there to exist a "natural" embedding class.
We argue that in any context like the one illustrated in our running example of
Bayesian model selection of binary phylogenetic trees there is no such
embedding
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
Likelihood Equations and Scattering Amplitudes
We relate scattering amplitudes in particle physics to maximum likelihood
estimation for discrete models in algebraic statistics. The scattering
potential plays the role of the log-likelihood function, and its critical
points are solutions to rational function equations. We study the ML degree of
low-rank tensor models in statistics, and we revisit physical theories proposed
by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical
algebraic geometry are employed to compute and certify critical points. We also
discuss positive models and how to compute their string amplitudes.Comment: 18 page
A Parsimonious Tour of Bayesian Model Uncertainty
Modern statistical software and machine learning libraries are enabling
semi-automated statistical inference. Within this context, it appears easier
and easier to try and fit many models to the data at hand, reversing thereby
the Fisherian way of conducting science by collecting data after the scientific
hypothesis (and hence the model) has been determined. The renewed goal of the
statistician becomes to help the practitioner choose within such large and
heterogeneous families of models, a task known as model selection. The Bayesian
paradigm offers a systematized way of assessing this problem. This approach,
launched by Harold Jeffreys in his 1935 book Theory of Probability, has
witnessed a remarkable evolution in the last decades, that has brought about
several new theoretical and methodological advances. Some of these recent
developments are the focus of this survey, which tries to present a unifying
perspective on work carried out by different communities. In particular, we
focus on non-asymptotic out-of-sample performance of Bayesian model selection
and averaging techniques, and draw connections with penalized maximum
likelihood. We also describe recent extensions to wider classes of
probabilistic frameworks including high-dimensional, unidentifiable, or
likelihood-free models
Model Selection for Stochastic Block Models
As a flexible representation for complex systems, networks (graphs) model entities and their interactions as nodes and edges. In many real-world networks, nodes divide naturally into functional communities, where nodes in the same group connect to the rest of the network in similar ways. Discovering such communities is an important part of modeling networks, as community structure offers clues to the processes which generated the graph. The stochastic block model is a popular network model based on community structures. It splits nodes into blocks, within which all nodes are stochastically equivalent in terms of how they connect to the rest of the network. As a generative model, it has a well-defined likelihood function with consistent parameter estimates. It is also highly flexible, capable of modeling a wide variety of community structures, including degree specific and overlapping communities. Performance of different block models vary under different scenarios. Picking the right model is crucial for successful network modeling. A good model choice should balance the trade-off between complexity and fit. The task of model selection is to automatically choose such a model given the data and the inference task. As a problem of wide interest, numerous statistical model selection techniques have been developed for classic independent data. Unfortunately, it has been a common mistake to use these techniques in block models without rigorous examinations of their derivations, ignoring the fact that some of the fundamental assumptions has been violated by moving into the domain of relational data sets such as networks. In this dissertation, I thoroughly exam the literature of statistical model selection techniques, including both Frequentist and Bayesian approaches. My goal is to develop principled statistical model selection criteria for block models by adapting classic methods for network data. I do this by running bootstrapping simulations with an efficient algorithm, and correcting classic model selection theories for block models based on the simulation data. The new model selection methods are verified by both synthetic and real world data sets