Imprecise Markov Semigroups and their Ergodicity

Abstract

We introduce the concept of an imprecise Markov semigroup \(\mathbf Q\). It is a tool that allows us to represent ambiguity around both the transition probabilities and the invariant measure of a continuous-time Markov process via a collection of Markov semigroups, each associated with a (possibly different) Markov process. We use techniques from topology, geometry, and probability to analyze ergodic limits under model uncertainty encoded by \(\mathbf Q\). We establish long-term bounds that are uniform in the initial state and identify regimes in which the imprecision in these bounds collapses asymptotically. Our results are proved in progressively more general settings. We first assume that \(\mathbf Q\) is compact and that the state space is Euclidean or a Riemannian manifold, working with a fixed bounded observable. We then allow the state space to be standard Borel, while keeping \(\mathbf Q\) compact and the observable fixed. Finally, we drop compactness and work on Polish metric spaces of finite diameter, where we treat arbitrary bounded Lipschitz observables. The importance of our findings for the fields of artificial intelligence and computer vision is also discussed at a high level; In particular, in the study of how the probability of an output evolves over time as we perturb the input of a convolutional autoencoder.

Figures (3)

• Figure 1: In this figure, we provide a visual representation of the functioning of a convolutional autoencoder in an image classification problem. The input image of a pug is mapped by encoding function $\phi_\text{enc}$ to the surface area $E_\text{dog} \in\mathscr{E}$ corresponding to the feature "dog". Such a projection is then mapped by decoding function $\phi_\text{dec}$ to the probability space over the labels. In this simple example the model gives a high probability to the first label, i.e. "dog".
• Figure 2: In this figure, we represent the smooth transition between a dog and a cat pictures. Moving a cursor between the pictures causes a walk from the "dog portion" $E_\text{dog}$ of the manifold's surface $\mathcal{S}(E)$ to the "cat portion" $E_\text{cat}$, depicted as a purple smooth curve. The results we present in this paper are useful to determine the behavior of a similar walk that is not deterministic, when it is not possible to determine exactly transition probabilities (and the invariant measure, when it exists), and when we are interested e.g. in the evolution of the probability of the first label, $y_1=\text{"dog"}$, captured by the functional $z \mapsto \tilde{f}(z)=e_1^\top \phi_\text{dec}(z)$.
• Figure 3: In this figure, $M$ is a Riemannian manifold in $\mathbb{R}^3$, $x\in M$ is an element of the manifold, and $T_xM$ is the tangent space of $M$ at $x$ (the set of all tangent vectors to all smooth curves on $M$ passing through $x$). $\gamma(t)$ is a $\mathcal{C}^1$ curve passing through $x$, and $v\equiv v(x)=(v^i(x))_{i\in\{1,2,3\}}$ is a vector field.

Theorems & Definitions (62)

• Definition 1: Good Measurable Space bakry
• Definition 2: Imprecise Markov Semigroup (IMSG)
• Remark 1: Antisymmetry and Identifiability
• Definition 3: Up/Down-Directed Subsets of $\mathbf{Q}_{\tilde{f}}$
• Definition 4: Dedekind-Closed Subsets of $\mathbf{Q}_{\tilde{f}}$
• Definition 5: Incomparability, Diversity, and $\preceq^\text{part}_{\tilde{f}}$-Width of $\mathbf{Q}_{\tilde{f}}$
• Definition 6: $\preceq^\text{part}_{\tilde{f}}$-Compatible Topology wolk1958order
• Remark 2: Interval Topology vs $\preceq$-Compatible Topologies
• Lemma 7: Topological Properties of $\mathbf{Q}_{\tilde{f}}$
• proof