Discussion and Conclusion

A traditional Markov network was applied for statically determining the status of the go board in [2]. Games played by people were used as data. Humans play the game better, but still, this approach has an important downside. The data contains only one possible future for each board position whereas a computer player can produce many possible futures. At the learning stage, all those futures can be used together for the computational price of one. Also, stones that are provably determined to be captured under optimal play (dead), might still be useful: By threathening to revive them, the player can gain elsewhere. When data is gathered with unoptimal play, the stones are marked as not quite dead, which might be desirable.

NRMN includes a probabilistic model only for the attributes and not for the logical relations. Link uncertainty means that one models the possibility of a certain relation to exist or not. Actually one can model link uncertainty using just the proposed methodology. All the uncertain relations are assumed to be logically true and an additional binary attribute is included to mark whether the link exists or not. One only needs to take into account that when this binary attribute gets the value zero, the dependencies between the other attributes are not modelled. Also, time series data can be represented using relations $\mathrm{obs}(T)$ for the observations at time $T$ and $\mathrm{ensues}(T1,T2)$ to denote that the time indices $T1$ and $T2$ are adjacent. These two examples give light to the generality of the proposed method. In [12], HNFA is augmented with a variance model. Modelling variances would be important also in the NRMN setting, because then each expert would produce an estimate of its accuracy and thus implicitly a weight compared to other experts. In [9], relational Markov networks were constructed to be discriminative so that the model is specialised to classification. The same could be applied here.

Conclusion: A model was proposed for data containing both relations and nonlinear dependencies. The model was built by combining two state-of-the art probabilistic models, hierarchical nonlinear factor analysis and relational Markov networks by using the product-of-experts combination rule. Many simplifying assumptions were made, such as diagonality of the posterior covariance matrix, and separate learning of experts. Also, learning the model structure (the set of clique templates) was left as future work. Experiments with the game of go give promise for the proposed methodology.

Acknowledgements: The author thanks Kristian Kersting, Harri Valpola, Markus Harva, and Alexander Ilin for useful discussions and comments. This research has been funded by the Finnish Centre of Excellence Programme (2000-2005) under the project New Information Processing Principles, and by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This publication only reflects the author's views.