224,604 research outputs found
Inference for determinantal point processes without spectral knowledge
Determinantal point processes (DPPs) are point process models that naturally
encode diversity between the points of a given realization, through a positive
definite kernel . DPPs possess desirable properties, such as exact sampling
or analyticity of the moments, but learning the parameters of kernel
through likelihood-based inference is not straightforward. First, the kernel
that appears in the likelihood is not , but another kernel related to
through an often intractable spectral decomposition. This issue is
typically bypassed in machine learning by directly parametrizing the kernel
, at the price of some interpretability of the model parameters. We follow
this approach here. Second, the likelihood has an intractable normalizing
constant, which takes the form of a large determinant in the case of a DPP over
a finite set of objects, and the form of a Fredholm determinant in the case of
a DPP over a continuous domain. Our main contribution is to derive bounds on
the likelihood of a DPP, both for finite and continuous domains. Unlike
previous work, our bounds are cheap to evaluate since they do not rely on
approximating the spectrum of a large matrix or an operator. Through usual
arguments, these bounds thus yield cheap variational inference and moderately
expensive exact Markov chain Monte Carlo inference methods for DPPs
Graphical modeling of stochastic processes driven by correlated errors
We study a class of graphs that represent local independence structures in
stochastic processes allowing for correlated error processes. Several graphs
may encode the same local independencies and we characterize such equivalence
classes of graphs. In the worst case, the number of conditions in our
characterizations grows superpolynomially as a function of the size of the node
set in the graph. We show that deciding Markov equivalence is coNP-complete
which suggests that our characterizations cannot be improved upon
substantially. We prove a global Markov property in the case of a multivariate
Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.Comment: 43 page
Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes
The application of Stochastic Differential Equations (SDEs) to the analysis
of temporal data has attracted increasing attention, due to their ability to
describe complex dynamics with physically interpretable equations. In this
paper, we introduce a non-parametric method for estimating the drift and
diffusion terms of SDEs from a densely observed discrete time series. The use
of Gaussian processes as priors permits working directly in a function-space
view and thus the inference takes place directly in this space. To cope with
the computational complexity that requires the use of Gaussian processes, a
sparse Gaussian process approximation is provided. This approximation permits
the efficient computation of predictions for the drift and diffusion terms by
using a distribution over a small subset of pseudo-samples. The proposed method
has been validated using both simulated data and real data from economy and
paleoclimatology. The application of the method to real data demonstrates its
ability to capture the behaviour of complex systems
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