3,042 research outputs found
Patterns of Scalable Bayesian Inference
Datasets are growing not just in size but in complexity, creating a demand
for rich models and quantification of uncertainty. Bayesian methods are an
excellent fit for this demand, but scaling Bayesian inference is a challenge.
In response to this challenge, there has been considerable recent work based on
varying assumptions about model structure, underlying computational resources,
and the importance of asymptotic correctness. As a result, there is a zoo of
ideas with few clear overarching principles.
In this paper, we seek to identify unifying principles, patterns, and
intuitions for scaling Bayesian inference. We review existing work on utilizing
modern computing resources with both MCMC and variational approximation
techniques. From this taxonomy of ideas, we characterize the general principles
that have proven successful for designing scalable inference procedures and
comment on the path forward
Integral, mean and covariance of the simplex-truncated multivariate normal distribution
Compositional data, which is data consisting of fractions or probabilities,
is common in many fields including ecology, economics, physical science and
political science. If these data would otherwise be normally distributed, their
spread can be conveniently represented by a multivariate normal distribution
truncated to the non-negative space under a unit simplex. Here this
distribution is called the simplex-truncated multivariate normal distribution.
For calculations on truncated distributions, it is often useful to obtain rapid
estimates of their integral, mean and covariance; these quantities
characterising the truncated distribution will generally possess different
values to the corresponding non-truncated distribution. In this paper, three
different approaches that can estimate the integral, mean and covariance of any
simplex-truncated multivariate normal distribution are described and compared.
These three approaches are (1) naive rejection sampling, (2) a method described
by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross
algorithm with an analytical version of elliptical slice sampling, and (3) a
semi-analytical method that expresses the integral, mean and covariance in
terms of integrals of hyperrectangularly-truncated multivariate normal
distributions, the latter of which are readily computed in modern mathematical
and statistical packages. Strong agreement is demonstrated between all three
approaches, but the most computationally efficient approach depends strongly
both on implementation details and the dimension of the simplex-truncated
multivariate normal distribution. For computations in low-dimensional
distributions, the semi-analytical method is fast and thus should be
considered. As the dimension increases, the Gessner et al. method becomes the
only practically efficient approach of the methods tested here
Discriminating between a Stochastic Gravitational Wave Background and Instrument Noise
The detection of a stochastic background of gravitational waves could
significantly impact our understanding of the physical processes that shaped
the early Universe. The challenge lies in separating the cosmological signal
from other stochastic processes such as instrument noise and astrophysical
foregrounds. One approach is to build two or more detectors and cross correlate
their output, thereby enhancing the common gravitational wave signal relative
to the uncorrelated instrument noise. When only one detector is available, as
will likely be the case with the Laser Interferometer Space Antenna (LISA),
alternative analysis techniques must be developed. Here we show that models of
the noise and signal transfer functions can be used to tease apart the
gravitational and instrument noise contributions. We discuss the role of
gravitational wave insensitive "null channels" formed from particular
combinations of the time delay interferometry, and derive a new combination
that maintains this insensitivity for unequal arm length detectors. We show
that, in the absence of astrophysical foregrounds, LISA could detect signals
with energy densities as low as with just
one month of data. We describe an end-to-end Bayesian analysis pipeline that is
able to search for, characterize and assign confidence levels for the detection
of a stochastic gravitational wave background, and demonstrate the
effectiveness of this approach using simulated data from the third round of
Mock LISA Data Challenges.Comment: 10 Pages, 10 Figure
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