Md. Asif Uddin

    Chapter 2 VI.2

    Causal Graphs

    A causal graph encodes conditional independences, and d-separation reads them off the graph alone.

    A DAG encodes conditional independences, read off without touching the data.

    How this chapter is built

    M3Load-bearing

    The content is mathematics. Understanding is demonstrated by computation, not recall.

    basics3/11what the words mean
    concept2/2what to picture
    theory0/4why it works, and when it does not
    mathematics0/15derive it, then compute it
    practice0/9build it, break it, read the papers

    Five strands, not one. Mathematics is the spine; the other four are the body. A chapter cannot pay its way out of teaching with problems, nor out of problems with teaching.

    Before you start

    The problem

    If assumptions must be made, they should be written where they can be inspected. A graph is that notation, and it comes with an algorithm for deciding which associations it predicts.

    Permutation equivariance, and the repairIn the upper panel, self-attention alone maps a shuffled input to an identically shuffled output: reordering the tokens carries no information. In the lower panel the same tokens carry positional encodings, so a reordering produces a genuinely different output.attention aloneattention + positionthecatsatsatthecatthecatsatsatthecat+p1+p2+p3+p1+p2+p3the′cat′sat′sat′the′cat′the′cat′sat′cat′sat′the′same output, merely reordered — the model cannot tell the two inputs apartdifferent output — order is now information
    Fig. 2 — Permutation equivariance and its repair. Without positional encoding a reordered input yields a merely reordered output; with it, order becomes information.

    What this chapter covers

    • DAGs
    • Nodes
    • Edges
    • Paths
    • Ancestors
    • Descendants
    • Colliders
    • Confounders

    Apparatus

    The mathematics this chapter leans on, held in Book 0 so it can be assumed here without being taught here. Not a gate — follow a link when a step stops making sense.

    Graphs, vertices and edges 0.GR.01 · Walks, paths, cycles and connectivity 0.GR.03 · Conditional independence 0.PR.05 · Distributions, discrete and continuous 0.PR.01

    Notation

    • X (random)A random variable
    • ⫫Statistical independence

    Propositions

    Not yet written. The topics above are the plan for this chapter; each will become a proposition with its own figure.

    Worked problems

    0/5 problems0/4 variants0/10 exercisesowes 15 more

    Not yet written. At M3 this chapter owes 5 worked problems across 4 distinct variants, and 10 exercises, every one with a published solution. The build enforces that from the day the chapter is marked published.