... tree^3.1

We assume for now that the join tree is really a tree.

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... linear^3.2

Note that the data is assumed to be zero mean here to simplify the equations.

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... equivalent^3.3

By either ignoring the conditioned $x(t-1)$ or including a copy of the observation to the hidden state.

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...^4.1

The use of nonlinear models is nontrivial, like choosing a good kernel in kernel PCA.

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... variables.^5.1

Atoms may in general be nested. For example $\mathrm{knows}(X,\mathrm{mother}(X,Y)) \leftarrow \mathrm{mother}(X,Y)$ means that every person $X$ knows that she is the mother of $Y$ if that really is the case. Nested atoms are out of the scope of this thesis.

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... page^5.2

http://www.cs.bris.ac.uk/~ILPnet2/

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... PRMs.^6.1

LOHMMs are acyclic by definition since all directed edges point from past to future.

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... generalised.^7.1

The algorithmic improvements in nonlinear state-space models are ongoing.

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