1,200 research outputs found
Finite-Sample Analysis of Fixed-k Nearest Neighbor Density Functional Estimators
We provide finite-sample analysis of a general framework for using k-nearest
neighbor statistics to estimate functionals of a nonparametric continuous
probability density, including entropies and divergences. Rather than plugging
a consistent density estimate (which requires as the sample size
) into the functional of interest, the estimators we consider fix
k and perform a bias correction. This is more efficient computationally, and,
as we show in certain cases, statistically, leading to faster convergence
rates. Our framework unifies several previous estimators, for most of which
ours are the first finite sample guarantees.Comment: 16 pages, 0 figure
Efficient Estimation of Mutual Information for Strongly Dependent Variables
We demonstrate that a popular class of nonparametric mutual information (MI)
estimators based on k-nearest-neighbor graphs requires number of samples that
scales exponentially with the true MI. Consequently, accurate estimation of MI
between two strongly dependent variables is possible only for prohibitively
large sample size. This important yet overlooked shortcoming of the existing
estimators is due to their implicit reliance on local uniformity of the
underlying joint distribution. We introduce a new estimator that is robust to
local non-uniformity, works well with limited data, and is able to capture
relationship strengths over many orders of magnitude. We demonstrate the
superior performance of the proposed estimator on both synthetic and real-world
data.Comment: 13 pages, to appear in International Conference on Artificial
Intelligence and Statistics (AISTATS) 201
Quantifying information transfer and mediation along causal pathways in complex systems
Measures of information transfer have become a popular approach to analyze
interactions in complex systems such as the Earth or the human brain from
measured time series. Recent work has focused on causal definitions of
information transfer excluding effects of common drivers and indirect
influences. While the former clearly constitutes a spurious causality, the aim
of the present article is to develop measures quantifying different notions of
the strength of information transfer along indirect causal paths, based on
first reconstructing the multivariate causal network (\emph{Tigramite}
approach). Another class of novel measures quantifies to what extent different
intermediate processes on causal paths contribute to an interaction mechanism
to determine pathways of causal information transfer. A rigorous mathematical
framework allows for a clear information-theoretic interpretation that can also
be related to the underlying dynamics as proven for certain classes of
processes. Generally, however, estimates of information transfer remain hard to
interpret for nonlinearly intertwined complex systems. But, if experiments or
mathematical models are not available, measuring pathways of information
transfer within the causal dependency structure allows at least for an
abstraction of the dynamics. The measures are illustrated on a climatological
example to disentangle pathways of atmospheric flow over Europe.Comment: 20 pages, 6 figure
Scalable Hash-Based Estimation of Divergence Measures
We propose a scalable divergence estimation method based on hashing. Consider
two continuous random variables and whose densities have bounded
support. We consider a particular locality sensitive random hashing, and
consider the ratio of samples in each hash bin having non-zero numbers of Y
samples. We prove that the weighted average of these ratios over all of the
hash bins converges to f-divergences between the two samples sets. We show that
the proposed estimator is optimal in terms of both MSE rate and computational
complexity. We derive the MSE rates for two families of smooth functions; the
H\"{o}lder smoothness class and differentiable functions. In particular, it is
proved that if the density functions have bounded derivatives up to the order
, where is the dimension of samples, the optimal parametric MSE rate
of can be achieved. The computational complexity is shown to be
, which is optimal. To the best of our knowledge, this is the first
empirical divergence estimator that has optimal computational complexity and
achieves the optimal parametric MSE estimation rate.Comment: 11 pages, Proceedings of the 21st International Conference on
Artificial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spai
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