An Algebraic Framework for Hierarchical Probabilistic Abstraction (2502.21216v1)

Abstract: Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current approaches often distill detailed probabilistic data into higher-level summaries to support tractable and interpretable analyses, though they typically struggle to fully represent the relational and probabilistic hierarchies through single-layered abstractions. We introduce a hierarchical probabilistic abstraction framework aimed at addressing these challenges by extending a measure-theoretic foundation for hierarchical abstraction. The framework enables modular problem-solving via layered mappings, facilitating both detailed layer-specific analysis and a cohesive system-wide understanding. This approach bridges high-level conceptualization with low-level perceptual data, enhancing interpretability and allowing layered analysis. Our framework provides a robust foundation for abstraction analysis across AI subfields, particularly in aligning System 1 and System 2 thinking, thereby supporting the development of diverse abstraction methodologies.

• The paper introduces an algebraic framework extending measure-theoretic abstraction to multi-layered probabilistic models, enabling dual-level reasoning and enhancing interpretability.
• It defines a taxonomy of one-layered abstraction processes (direct, divergent, convergent) and introduces Hierarchical Probabilistic Abstraction Models (HPAMs) using DAGs.
• The framework is applicable to complex systems like educational models or disease progression, facilitating structured analysis despite potential computational complexity.

The paper introduces an algebraic framework for hierarchical probabilistic abstraction to address the challenges of designing effective abstraction methodologies for probabilistic models. The framework aims to enhance interpretability and enable layered analysis by bridging high-level conceptualization with low-level perceptual data.

Key concepts and contributions:

• The paper addresses the limitations of single-layered probabilistic abstractions in capturing the full relational and probabilistic hierarchy in complex systems.
• It extends the measure-theoretic approach to multi-layered settings, providing a structured foundation for abstraction in complex systems. The framework decomposes abstraction into layered mappings, supporting detailed, layer-specific analysis alongside a comprehensive system-level understanding. This enables dual-level reasoning, considering both individual transformations and probabilistic relationships at each layer, as well as cumulative effects across layers.
• The framework facilitates exploration of abstractions across AI subfields, notably in System 1 and System 2 cognition, by connecting high-level conceptual reasoning with low-level perceptual data.

The paper presents a motivating example of a personalized learning system modeled using a probabilistic relational model (PRM) for a university database $U$. The model incorporates constraints for a parameterized Bayesian network:

$$\text{Environmental Factors} \rightarrow \text{Cognitive Processes} \leftarrow \text{Learning Behaviors} \rightarrow \text{Educational Outcomes}$$

The example illustrates the limitations of a two-layer abstraction in capturing the complex interdependencies within educational systems, such as the relationships between Home Learning Environment (HLE), Parental Involvement (PI), Socioeconomic Status (SES), Cognitive Abilities, and Learning Styles.

The paper introduces a taxonomy of abstraction processes based on one-layered mappings, which are then used to build multi-layered hierarchies in a compositional manner. The abstraction processes are categorized based on structural characteristics and mapping nature:

1. Direct Abstraction (One-to-One):
   • Definition: Given a concrete probability space $(\Omega_{c}, \Sigma_{c}, \mu_{c})$ and an abstract probability space $(\Omega_{a}, \Sigma_{a}, \mu_{a})$, direct abstraction is facilitated by a bijective measurable function $\mathcal{A}: \Omega_{c} \rightarrow \Omega_{a}$.
   • The abstract measure $\mu_{a}$ is the pushforward of the concrete measure $\mu_{c}$ via $\mathcal{A}$, defined by $\mu_{a}(B) = \mu_{c}(\mathcal{A}^{-1}(B))$ for all $B \in \Sigma_{a}$.
2. Divergent Abstraction (One-to-Many):
   • Definition: Given a concrete probability space $(\Omega_{c}, \Sigma_{c}, \mu_{c})$, divergent abstraction maps this space to a collection of abstract probability spaces $$\{(\Omega_{a(i)}, \Sigma_{a(i)}, \mu_{a(i)})\}_{i \in I}$$ via measurable functions $$\{\mathcal{A}_i: \Omega_{c} \rightarrow \Omega_{a(i)}\}_{i \in I}$$.
   • A unified abstract measure $\mu^{'}$ on $\Sigma^{'}$ is constructed such that for any $E \in \Sigma^{'}$, $\mu^{'}(E)$ integrates or aggregates $\mu_{a(i)}(E \cap \Omega_{a(i)})$ for all $i \in I$, normalized to ensure $\mu^{'}$ is a probability measure.
3. Convergent Abstraction (Many-to-One):
   • Definition: Given a collection of concrete probability spaces $$\{(\Omega_{c(i)}, \Sigma_{c(i)}, \mu_{c(i)})\}_{i \in I}$$ and an abstract probability space $(\Omega_{a}, \Sigma_{a}, \mu_{a})$, convergent abstraction is achieved through measurable functions $$\{\mathcal{A}_i: \Omega_{c(i)} \rightarrow \Omega_{a}\}_{i \in I}$$.
   • The abstract measure $\mu_{a}$ integrates the measures from all concrete spaces, defined for any $B \in \Sigma_{a}$ as $$\mu_{a}(B) = \sum_{i \in I} \mu_{c(i)}(\mathcal{A}_i^{-1}(B))$$, with appropriate normalization to ensure $\mu_{a}$ remains a probability measure.

The paper introduces two types of Hierarchical Probabilistic Abstraction Models (HPAMs):

1. HPAM-DAG (Directed Acyclic Graphs): Uses a tree-like structure for systems with clear, sequential progressions and no cycles.
   • Definition: A Hierarchical Probabilistic Abstraction, denoted as $\mathcal{H}$, is defined as a triple $\left( \mathcal{V}, \mathcal{E}, \{\mathcal{P}_v\}_{v \in \mathcal{V} \right)$ where:
     • $\mathcal{V}$ is a set of vertices in a Directed Acyclic Graph (DAG), each corresponding to a distinct probabilistic space.
     • $$\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$$ is a set of directed edges in the DAG, each representing an abstraction mapping between probabilistic spaces.
     • $\{\mathcal{P}_v\}_{v \in \mathcal{V}}$ is a family of probabilistic spaces associated with vertices in $\mathcal{V}$.
   • For every directed edge $(v_i, v_j) \in \mathcal{E}$ within the DAG, there exists an abstraction mapping $$\mathcal{A}_{ij}: \Omega_{v_i} \rightarrow \Omega_{v_j}$$ that facilitates a transformation between probabilistic spaces $\mathcal{P}_{v_i}$ and $\mathcal{P}_{v_j}$. This mapping satisfies the condition:

     $$P_{v_j}(A_{v_j}) = P_{v_i}(\mathcal{A}_{ij}^{-1}(A_{v_j}))$$

2. HPAM-CD (Cyclical and Dynamic): Captures systems with feedback loops, cycles, and dynamic interactions, but is reserved for future work due to complexities.

The paper defines the Highest Possible Abstraction (HPoA) as the boundary for abstraction in probabilistic models. Beyond the HPoA, further abstraction risks obscuring critical probabilistic or relational details.

• Definition: Within a HPAM-DAG denoted as $\mathcal{H} = \left( \mathcal{V}, \mathcal{E}, \{\mathcal{P}_v\}_{v \in \mathcal{V} \right)$, HPoA is formalized as a specific probabilistic space $(\Omega_{HPoA}, \Sigma_{HPoA}, P_{HPoA})$ that meets the following criteria:

  • Preservation of Probabilistic Integrity:

    $$\forall A_{HPoA} \in \Sigma_{HPoA}, \exists A_{\mathcal{P}_v} \in \Sigma_{\mathcal{P}_v} \text{ for } \mathcal{P}_v \in \{\mathcal{P}_v\}_{v \in \mathcal{V}: \ P_{HPoA}(A_{HPoA}) = P_{\mathcal{P}_v}(\mathcal{A}_{v \rightarrow HPoA}^{-1}(A_{HPoA}))}$$

  • Maximal Generalization:

    $$\not\exists (\Omega', \Sigma', P') : \Sigma_{HPoA} \subsetneq \Sigma' \land \forall A' \in \Sigma', P'(A') \neq P_{HPoA}(\mathcal{A}_{HPoA \rightarrow '}^{-1}(A')).$$

The paper outlines two types of HPAM-DAGs: sequential abstraction and a more complex hybrid variant.

1. Sequential Abstraction:
   • Definition: Given a foundational probability space $(\Omega_0, \Sigma_0, P_0)$, sequential abstraction is characterized by a finite series of probability spaces $\{(\Omega_i, \Sigma_i, P_i)\}_{i=0}^n$, where each space $(\Omega_{i+1}, \Sigma_{i+1}, P_{i+1})$ is derived from $(\Omega_i, \Sigma_i, P_i)$ via an abstraction operation $\mathcal{A}_i$.
   • For each $i$, ranging from $1$ to $n$: $$\mathcal{A}_i: (\Omega_{i-1}, \Sigma_{i-1}, P_{i-1}) \rightarrow (\Omega_i, \Sigma_i, P_i)$$, under the stipulations that:
     • for every measurable set $A \in \Sigma_{i-1}$, the condition $P_{i-1}(A) = P_i(\mathcal{A}_i(A))$ holds true (Preservation of Probability Mass), and
     • there is no $\sigma$-algebra smaller than $\Sigma_i$, denoted as $\Sigma' \subsetneq \Sigma_i$, for which the preservation of probability mass criteria remains valid (Minimality).
2. Hybrid HPAMs:
   • Hybrid HPAM-DAGs amalgamate sequential, divergent, and convergent abstraction methodologies within hierarchical probabilistic modeling.

To exemplify the Hybrid HPAM, the paper describes a model for the dynamic nature of complex systems like Alzheimer's disease. It begins with a foundational concrete space $(\Omega_0, \Sigma_0, P_0)$, where $\Omega_0$ represents the population at risk for Alzheimer's disease, $\Sigma_0$ includes measurable events indicating risk factors for Alzheimer's, and $P_0$ quantifies the initial distribution of these risk factors.

1. Sequential Abstraction: The abstraction transitions the model from broad risk factors to specific biological markers indicative of Alzheimer's:

   $$\mathcal{A}_1: (\Omega_0, \Sigma_0, P_0) \rightarrow (\Omega_1, \Sigma_1, P_1)$$

2. Divergent Abstraction: From this point, the model branches into distinct intervention pathways:

   $$\mathcal{A}_{3,i}: (\Omega_2, \Sigma_2, P_2) \rightarrow (\Omega_{3,i}, \Sigma_{3,i}, P_{3,i}), \quad i \in \{1, 2\}$$

3. Convergent Abstraction: The insights from these divergent paths are then synthesized into a unified model:

   $$\mathcal{A}_4: (\Omega_{3,1} \cup \Omega_{3,2}, \Sigma_{3,1} \cup \Sigma_{3,2}, P_{3,1} \oplus P_{3,2}) \rightarrow (\Omega_4, \Sigma_4, P_4)$$

The paper concludes by acknowledging the limitations of the framework, including increased computational complexity, the trade-off between detail and simplicity, and the reliance on data quality. Future work will focus on refining computational strategies, investigating additional taxonomies of abstraction, and conducting extensive testing across diverse domains.