**Preamble**

Fractal Tree (Wikipedia) |

**Understanding weighted Directed Acyclic Graphs (wDAGs) as causal data structure **

We first define what is a directions and weights. Providing a notational definition via tuples of objects.

__ Definition (wDAG)__: A weighted Directed Acyclic Graph (wDAG) $\mathscr{G_{c}}$ is defined as set of

*ordered*triplets of weights and connected random variables, such that, $k$th triplet $(w_{k}, x_{i}, x_{j})$ where by $w_{k} \in \mathbb{R}$ is the weight, an effect size, between two variates that $x_{i}$ effects $x_{j}$. There are constraints :

(i) No cyclic effects can be defined, necessarily $x_{i}$ can not be equal to $ x_{j}$.

(ii) If there is a definition, $(w_{k}, x_{i}, x_{j})$ the reverse can't be defined, i.e., so that $(w_{k}, x_{j}, x_{i})$ does not exist.

(iii) No two causal effects sizes can't be exactly equal, $w_{k}$ can not be equal to $w_{l}$, from the same causal variable, meaning no simultaneous events caused by the same random variable. This prevents ambiguity of ordering and random tie-breaks are unnatural.

This definition is practical and do not introduce any graph theory jargon. We left the sizes of indices as an exercise.

**Inducing Causal Order via wDAGs**

By the very definition of wDAGs, the power of this definition is one can construct causal ordering.

* Definition (Causal Ordering from wDAG)*: Given $\mathscr{G_{c}}$, we can construct causal ordering among random variates $O(i)$ for $x_{i}$ using directionality and weights from $\mathscr{G_{c}}$:

(i) if there exist a triplet $(w_{k}, x_{i}, x_{j})$, then ordering $x_{j} \succ x_{i}$, implies $x_{j}$ occurred before $x_{j}$, or cause of $x_{i}$ was $x_{j}$

(ii) if there are two or more triplets having the same first variates, ordering is induces by the effect size $w_{k}$ among them.

To provide a simple example, let's say we formed a wDAG, $\mathscr{G_{c}} = \{ (0.1, x_{1}, x_{2}),(0.2, x_{1}, x_{3}), (1.1, x_{2}, x_{4}) \}$ then the following causal ordering is established $x_{1} \succ x_{3} \succ x_{2} \succ x_{4}$, note the ordering of $x_{3}$ that took precedence on $x_{2}$ due to its weight.

**Why LLMs with causal ordering are so successful?**

Probably not very well spelled property of LLMs are having causal layers with deep learning elevating their ability to capture causal ordering in natural language so well, not only sequence. This is still in infancy from research perspective as LLMs are biologically not plausible engineered software systems act as lossy knowledge compressors, lossy part usually identified as hallucination.

**Conclusion**

We introduce basic definition of wDAGs without heavy graph theory jargon and provide hints on why causal ordering with wDAGs has an immense contribution in constructing useful LLMs.

**Further reading**

- looper : Causality Resource List. [link]
- Pearl et. al. Causal Inference in Statistics [link]
- Shimizu et. al., A Linear Non-Gaussian Acyclic Model for Causal Discovery [link]
- Hamad, et. al, Dilated causal convolution with multi-head self attention for sensor human activity recognition [link].
- Lui et. al DecBERT: Enhancing the Language Understanding of BERT with Causal Attention Masks [link]

*Postscript Notes**Last Update 18 May 2024*

- Transformers mathematically mimics wDAGs, i.e., induce causal ordering among distributed representations via “Attention” i.e., weighting inputs and linking them to different types of output sequences.
- Directed Acyclic Graphs (DAGs), are actually build implicitly by transformer architectures. This lead to causal analysis and preserving ordering of most meaningful word embeddings in an empirical level, i.e., heuristic causality bound by data-driven model with inductive bias of the modeller (LLM trainer). Striking enough, it is possible to express these relationships using triplet notation without Graph Theory, and that’s what attention layer store rather than graph.
- The primary driver in this is not that it can only place an
*attention*to certain portions of the embedded vector space for approximating the separation of meaning. These data processing instructions actually builds a causal mappings among sub-spaces, i.e., a causal ordering among embedding sub-spaces. In short,.__transformers are heuristically executing causal discovery__