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Let $(\Omega, E)$ be a measure space. An $n$-dimensional statistical model is then a tuple $(\Theta, \mathcal{M}, \Phi)$ where $\Theta \subseteq \mathbb{R}^n$ open, $\mathcal{M} = \{p_\theta := p(\cdot | \theta), \theta \in \Theta\}$ is the set of parameterized probability distributions on $\Omega$ and the mapping $\Phi: \Theta \to \mathcal{M}$ via $\theta \mapsto p(\cdot | \theta)$ is injective.$^1$

In this context, one refers to $\mathcal{M}$ as a statistical manifold, as the mapping $\Phi$ serves as a coordinate system for $\mathcal{M}$.

Question: Can anyone give me an example of a statistical manifold with non-trivial topology? If this is not possible, can you give an explanation for why this is.

Many thanks!


$^1$ This description is taken from here.

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You can always restrict your statistical manifold of interest to a submanifold with non-trivial topology, which would still match your definition.

For probability families of the usual type, their naturally associated statistical manifolds have trivial topology, but this does not have to be the case for more complicated models. If the parameters of your statistical model need to satisfy some constraints this may confer your manifold with interesting topology.

In fact, some fundamental aspects of information may be captured topologically and information topology is a growing field of research. For details see https://arxiv.org/abs/1409.6203 or https://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/ which argues why information is fundamentally topological.

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