My main research interests are in machine learning, probabilistic graphical models and discrete optimisation using integer programming. I also work on statistical relational learning and, occasionally, philosophy of probability. Here are some possible topics for a PhD. Let me know if you're interested!

- Scaling up Bayesian network learning. We live in an
age of 'big data' and our machine learning algorithms must be able to
learn from large datasets. One important machine learning task is
learning 'Bayesian networks' (BNs) from data. BNs provide a graphical
way of seeing patterns in the data and also allow us to make
predictions about the future.
At York we have developed the GOBNILP Bayesian
network learner. GOBNILP (typically) does
**exact**learning where an attempt to find the best possible network (according to some score) is found. Since BN learning is known to be a 'hard' problem (NP-hard to be formal) it will always be possible to find big datasets where this attempt will fail. But this should not stop us attempting to scale up BN learning as much as possible! The PhD topic here is to push the boundaries of BN learning to cope with ever bigger datasets. - Beyond Bayesian networks . I have been working on learning directed graphical models ('Bayesian networks') from complete discrete data. Extending this approach to other graphical model learning problems is an exciting research area. Problems include: learning Gaussian graphical models (decomposable or unrestricted), chain event graphs, non-graphical log-linear models (probably restricted to hierarchical ones), etc, etc.
- From optimisation to integration. In Bayesian statistics finding the most probable model is useful and I have applied integer programming to solve this optimisation problem (for directed graphical models with complete discrete data). However a full Bayesian approach requires consideration of the entire posterior distribution over models. Typically one wants to compute some marginal posterior quantity (e.g. the posterior probability that some edge exists in a graphical model). This requires computing weighted sums (discrete integration) over a very large set (eg the set of all acyclic digraphs). Work by Ermon and colleagues has shown that one can get good approximations to this sum by repeatedly solving optimisation problems with random constraints. This area merits further research. In particular it would be useful to compare it to MCMC approaches to the same problem.
- Mixed integer programming for product space approaches. Integer programming has been used for Bayesian model selection when it has been possible to analytically 'integrate away' model parameters. This is, of course, a big restriction. It would be useful to look into applying mixed integer programming (where both discrete and continuous variables are used) to sampling from a 'product space' encoding both models (discrete) and parameters (continuous). An extension to the 'random constraint' approach mentioned above should allow such sampling to be reduced to repeated optimisations.

To undertake research in these areas it is necessary to have a strong mathematical background.

You will be a member of both the Artificial Intelligence Group and the York Centre for Complex Systems Analysis, and will be working in a top-rated department. In 2014, the Department of Computer Science ranked joint 7th with Oxford (among 89 CS departments in the country) for the quality of our research.

As a member of the York Centre for Complex Systems Analysis,
I work in the Ron Cooke Hub, a new,
purpose-built building in the city of York,
one of the most historic, picturesque and safe cities in the
United Kingdom. The Hub hosts several world-class research
groups and a growing number of startups.

Please have a look at this page for information regarding available scholarships/funding.