44,501 research outputs found
Robust estimation of risks from small samples
Data-driven risk analysis involves the inference of probability distributions
from measured or simulated data. In the case of a highly reliable system, such
as the electricity grid, the amount of relevant data is often exceedingly
limited, but the impact of estimation errors may be very large. This paper
presents a robust nonparametric Bayesian method to infer possible underlying
distributions. The method obtains rigorous error bounds even for small samples
taken from ill-behaved distributions. The approach taken has a natural
interpretation in terms of the intervals between ordered observations, where
allocation of probability mass across intervals is well-specified, but the
location of that mass within each interval is unconstrained. This formulation
gives rise to a straightforward computational resampling method: Bayesian
Interval Sampling. In a comparison with common alternative approaches, it is
shown to satisfy strict error bounds even for ill-behaved distributions.Comment: 13 pages, 3 figures; supplementary information provided. A revised
version of this manuscript has been accepted for publication in Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering
Science
Comparing stochastic design decision belief models : pointwise versus interval probabilities.
Decision support systems can either directly support a product designer or support an agent operating within a multi-agent system (MAS). Stochastic based decision support systems require an underlying belief model that encodes domain knowledge. The underlying supporting belief model has traditionally been a probability distribution function (PDF) which uses pointwise probabilities for all possible outcomes. This can present a challenge during the knowledge elicitation process. To overcome this, it is proposed to test the performance of a credal set belief model. Credal sets (sometimes also referred to as p-boxes) use interval probabilities rather than pointwise probabilities and therefore are more easier to elicit from domain experts. The PDF and credal set belief models are compared using a design domain MAS which is able to learn, and thereby refine, the belief model based on its experience. The outcome of the experiment illustrates that there is no significant difference between the PDF based and credal set based belief models in the performance of the MAS
Scattering statistics of rock outcrops: Model-data comparisons and Bayesian inference using mixture distributions
The probability density function of the acoustic field amplitude scattered by
the seafloor was measured in a rocky environment off the coast of Norway using
a synthetic aperture sonar system, and is reported here in terms of the
probability of false alarm. Interpretation of the measurements focused on
finding appropriate class of statistical models (single versus two-component
mixture models), and on appropriate models within these two classes. It was
found that two-component mixture models performed better than single models.
The two mixture models that performed the best (and had a basis in the physics
of scattering) were a mixture between two K distributions, and a mixture
between a Rayleigh and generalized Pareto distribution. Bayes' theorem was used
to estimate the probability density function of the mixture model parameters.
It was found that the K-K mixture exhibits significant correlation between its
parameters. The mixture between the Rayleigh and generalized Pareto
distributions also had significant parameter correlation, but also contained
multiple modes. We conclude that the mixture between two K distributions is the
most applicable to this dataset.Comment: 15 pages, 7 figures, Accepted to the Journal of the Acoustical
Society of Americ
What we talk about when we talk about numbers
In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism
- …