Blurred Lines
Welcome to the second post of my series that aims to delve into the mechanics of narratives. In this post, we'll informally tackle the trade-off between abstracting and keeping a precise causal picture.
Our stage is set with an agent who desires to communicate about a specific structure: a Directed Acyclic Graph (DAG). In this context, the graph is ordered by time, and arrows symbolize causation. This makes the DAG a natural model for representing causal relationships in a temporal context.
Let's take a concrete example to illustrate what this graph could represent. Suppose our DAG is a historical account of certain events leading up to a war. The vertices in the graph are the events, and the arrows represent causality - event A led to event B, which in turn caused event C, and so forth.
Abstractions can be seen as groupings of the DAG into a new graph. Take our war history example: We might want to group together several minor skirmishes into a single node representing a larger conflict, or various diplomatic events into a single vertex representing an overall peace effort.
However, when we create these abstractions, we also necessarily blur or smear some of the original graph's details. Notably, our simplified graph resulting from the abstraction might not even be directed, and it might not be acyclic. Ideally, we'd want it to be both, to preserve the chronological and causal nature of the events represented.
Let’s introduce "Causality Preserving Abstractions" (CPAs). A CPA is an abstraction that maintains the direction of the arrows between grouped subgraphs in the DAG. We want this because it allows us to maintain the causal relationships between the abstracted events. In our war history example, a CPA would allow us to group minor skirmishes into larger conflicts while still preserving the overall chain of causality that led to the war.
To quantify the extent to which an abstraction preserves causality, we will introduce the measure "clarity." Clarity could be the fraction of causations that got smeared into being correlations only, weighted by how badly they were smeared. A high clarity indicates a good approximation of causality, while a low clarity represents a significant loss of causal information. In other words, clarity should measure how close an abstraction is to being a CPA.
To illustrate why arriving at abstractions might not always be CPAs, consider the case where the vertices of the DAG are not just isolated events but points in a high-dimensional embedding space - part of the agent's world model. The process of clustering these points into a hierarchical abstraction structure could come at a trade-off with clarity. We might introduce a quantitative measure, such as "fidelity," that balances the clarity of the abstraction with the quality of the clustering in the high-dimensional space. In our war history example, we might group conflicts by metrics such as size, death tolls and weapons used, and end up with a grouping of the conflicts that loses causal clarity but may permit some other useful classification of the skirmishes.
Just as we have established the trade-off between clarity and fidelity in the domain of abstractions, a similar trade-off exists in the realm of information retrieval and machine learning, known as the precision-recall trade-off. Precision is a measure of how relevant the results we get are, while recall measures how many of the relevant results we have managed to find.
Just like the precision-recall tradeoff, there's a balance to be struck between clarity and fidelity in our abstractions. High clarity (analogous to precision) means our abstractions preserve causality very well, and the arrows (causations) are well-defined; however, this might come at the cost of fidelity - we might not be grouping together nodes that are conceptually similar, or we might oversimplify the representation of our world model. On the other hand, high fidelity (analogous to recall) means we are making abstractions that maintain the nuanced, high-dimensional structure of our world model, but this might blur the causal relationships in our model, as the directionality and acyclicity of the arrows (causations) can become compromised. The challenge, therefore, is to find an optimal balance that suits the needs of the agent and the demands of its environment, just as in the case of precision and recall.
In the next post, I will re-introduce the terms Abstraction, Smearing, CPA, Clarity, Quality and Fidelity rigorously and take a look at how Vapnik-Chervonenkis Theory can be applied here.
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Stay abstract, folks :)