In our work on Probabilistic Ontology-Driven Multi-Agent Systems (pODMAS), entities, relationships, and interactions are represented as Gaussian Mixture Models (GMMs) within abstract embedding spaces. These GMMs serve as probabilistic representations of entities at specific points in time, capturing their properties and relationships within a multi-dimensional framework.
The Challenge: Evolving Entities Over Time
One of the key challenges in this framework is how to model the temporal evolution of entities. Since a GMM is based on a set of training samples, it represents a snapshot of an entity at a given point. Over time, new information and interactions will shift the understanding of that entity. The question becomes: how do we update these GMMs to reflect evolving realities?
Evolving GMMs with New Data
1. Online Learning
The most straightforward approach is to use online learning, where the GMM is updated incrementally as new data arrives. Techniques like Incremental Expectation-Maximization (EM) allow the GMM to evolve without retraining from scratch. This method continuously adjusts the parameters, allowing the model to adapt to changes in real-time.
2. Bayesian Updating
An alternative is Bayesian updating, where GMM parameters are treated as random variables with prior distributions. As new data is collected, Bayesian inference updates the GMM, refining its representation. This approach offers a more sophisticated treatment of uncertainty and change over time.
3. Temporal Modeling with Layered Approaches
By using temporal models like Hidden Markov Models (HMMs) or Recurrent Neural Networks (RNNs) in combination with GMMs, we can capture the sequence of data changes over time. These models can help predict future states, allowing the GMM to evolve with a built-in notion of time. Bitemporal Representation: Tracking Two Dimensions of Time
To rigorously account for time, we can incorporate bitemporality into the system. This involves tracking two dimensions of time:
- Valid Time: Embedded in the GMM, this represents when an entity’s state is true in the real world.
- Transaction Time: Linked to online learning iterations, this dimension tracks when the system learned or updated its knowledge about an entity.
With this structure, GMMs evolve in a dual-temporal framework, where the valid time captures external changes and the transaction time logs internal updates. This allows the system to maintain a temporal audit trail and retrieve information across different time periods. Managing GMM Complexity
A potential issue with updating GMMs is model complexity. While one could add a new GMM component with each online update, this would lead to overfitting and inefficiency. Instead, we advocate for adaptive component management:
- Selective Integration: New components are added only when new data diverges significantly from existing components.
- Component Merging and Splitting: Over time, GMM components can be merged or split based on evolving data, ensuring the model adapts without growing unnecessarily complex.
- Online Learning Algorithms: Techniques like Online EM update existing components rather than creating new ones for every update.
This approach ensures that the GMM stays responsive to meaningful changes while avoiding overreaction to noise, maintaining a balance between complexity and adaptability. Conclusion: Adaptive Evolution with Probabilistic Models
Our pODMAS system provides a robust framework for representing and evolving entities over time. By embedding valid time in GMMs and using online learning for transaction time, the system captures both external realities and its internal knowledge evolution. Adaptive management of GMM components ensures that the model remains computationally efficient while continuously learning from new data. As we continue to develop this framework, its potential applications range from economic modeling to dynamic ecosystems, making it a versatile tool for representing complex, evolving systems.
