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Uncertainty in Neural Networks; Bayesian Ensembles, Priors & Prediction Intervals
The breakout success of deep neural networks (NNs) in the 2010's marked a new era in the quest to build artificial intelligence (AI). With NNs as the building block of these systems, excellent performance has been achieved on narrow, well-defined tasks where large amounts of data are available.
However, these systems lack certain capabilities that are important for broad use in real-world applications. One such capability is the communication of uncertainty in a NN's predictions and decisions. In applications such as healthcare recommendation or heavy machinery prognostics, it is vital that AI systems be aware of and express their uncertainty β this creates safer, more cautious, and ultimately more useful systems.
This thesis explores how to engineer NNs to communicate robust uncertainty estimates on their predictions, whilst minimising the impact on usability. One way to encourage uncertainty estimates to be robust is to adopt the Bayesian framework, which offers a principled approach to handling uncertainty. Two of the major contributions in this thesis relate to Bayesian NNs (BNNs).
Specifying appropriate priors is an important step in any Bayesian model, yet it is not clear how to do this in BNNs. The first contribution shows that the connection between BNNs and Gaussian Processes (GPs) provides an effective lens to study BNN priors. NN architectures are derived which mirror the combining of GP kernels to create priors tailored to a task.
The second major contribution is a novel way to perform approximate Bayesian inference in BNNs using a modified version of ensembling. Novel analysis improves an understanding of a technique known as randomised MAP sampling. It's shown this is particularly effective when strong correlations exist between parameters, making it well suited to NNs.
The third major contribution of the thesis is a non-Bayesian technique that trains a NN to directly output prediction intervals for regression tasks through a tailored objective function. This advances over related works that were incompatible with gradient descent, and ignored one source of uncertainty.EPSRC, Alan Turing Institut
Constraining the orbits of sub-stellar companions imaged over short orbital arcs
Imaging a star's companion at multiple epochs over a short orbital arc
provides only four of the six coordinates required for a unique orbital
solution. Probability distributions of possible solutions are commonly
generated by Monte Carlo (MCMC) analysis, but these are biased by priors and
may not probe the full parameter space. We suggest alternative methods to
characterise possible orbits, which compliment the MCMC technique. Firstly the
allowed ranges of orbital elements are prior-independent, and we provide means
to calculate these ranges without numerical analyses. Hence several interesting
constraints (including whether a companion even can be bound, its minimum
possible semi-major axis and its minimum eccentricity) may be quickly computed
using our relations as soon as orbital motion is detected. We also suggest an
alternative to posterior probability distributions as a means to present
possible orbital elements, namely contour plots of elements as functions of
line of sight coordinates. These plots are prior-independent, readily show
degeneracies between elements and allow readers to extract orbital solutions
themselves. This approach is particularly useful when there are other
constraints on the geometry, for example if a companion's orbit is assumed to
be aligned with a disc. As examples we apply our methods to several imaged
sub-stellar companions including Fomalhaut b, and for the latter object we show
how different origin hypotheses affect its possible orbital solutions. We also
examine visual companions of A- and G-type main sequence stars in the
Washington Double Star Catalogue, and show that per cent must be
unbound.Comment: Accepted for publication in MNRA
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