Md. Asif Uddin

    Chapter 5 VI.5

    Counterfactuals

    A counterfactual asks about an individual outcome that was never observed, and no experiment can supply it.

    An individual outcome that was never observed.

    How this chapter is built

    M3Load-bearing

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

    basics1/11what 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

    Effects on populations are identifiable; effects on a person are not. The distinction is formal, it has a name, and it bounds what any method in this Book can claim about an individual.

    Two scenes, one projectionA camera centre with two objects along the same ray: a small one nearby and a large one far away. Both fill the same region of the image, so a single photograph cannot separate size from distance. Monocular depth is therefore predicted only up to an unknown scale.the same pixelscameranear, smallfar, largeimageBoth objects subtend the same angle, so both occupy the same pixels.Monocular depth models predict relative or scale-invariant depth — a metric claim needs stereo,a known object size, or a sensor that measures distance directly.
    Fig. 5 — Two scenes projecting to identical pixels. One photograph cannot separate size from distance.

    What this chapter covers

    • Potential outcomes
    • Counterfactual worlds
    • Individual treatment effects
    • Causal effect

    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.

    Expectation 0.PR.02 · Bayes' rule 0.PR.04 · Estimators, bias and variance 0.ST.01

    Notation

    • X (random)A random variable
    • 𝔼Expectation
    • do(·)The intervention operator
    • θ̂An estimate, as against the quantity it estimates

    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.