# HCGM Toy Teaching Path > Status: teaching entrance seed v0.1 > Scope: first reader path through the current five HCGM toy assets ## Purpose This file turns the current toy folder into a minimal teaching entrance. It is not a full tutorial book. It only answers: ```text If a reader has one hour, in what order should they inspect the current toys so they can understand why HCGM is not ordinary prediction and not merely a causal graph slogan? ``` For comparator-focused teaching, read: ```text TEACHING_CONTRASTS.md ``` `TEACHING_PATH.md` gives the first-reader order. `TEACHING_CONTRASTS.md` gives the reason each toy helps explain HCGM as a candidate first-choice framework relative to other modeling grammars. ## Reader Assumption The first reader may know basic statistics or machine learning, but still confuses: ```text observational prediction interventional prediction same-unit counterfactual prediction heterogeneous unit-level mechanisms ``` ## Path Overview ```text 1. HCGM regression worked example 2. Cauchy-affine predictive distribution 3. Simpson paradox 4. Berkson paradox 5. Surrogate-marker paradox ``` The order is intentional: ```text first learn the HCGM spine then see why the Cauchy + affine seed is tractable then inspect three ways observational evidence can mislead mechanism reasoning ``` ## Teaching In The Shared Claim Spine The teaching path should make the shared HCGM-first claim concrete: ```text HCGM is not just a new predictor; it is a grammar that separates factual evidence, target condition, unit belief, personalized mechanism, event noise, and Reduction. ``` Each toy should teach one part of that separation: ```text worked example -> factual evidence vs target condition Cauchy toy -> assumptions that make Reduction inspectable Simpson toy -> aggregate trend vs target unit distribution Berkson toy -> observed sample vs target population surrogate toy -> proxy path vs outcome mechanism ``` This is still a first-reader path, not a full tutorial chapter. ## 1. HCGM Regression Worked Example Folder: ```text toy/hcgm-regression-worked-example/ ``` Read first because it directly teaches the HCGM spine: ```text Abduction -> Prediction -> Reduction ``` What to notice: - Each unit has a latent representation type. - Evidence updates a belief over unit representation. - Target input \(x\) enters the personalized mechanism. - Reduction integrates over unit belief and event noise. Reader checkpoint: ```text Can you explain why using f(x,E;u) at a target x may be do(x)-style even inside a regression prediction problem? ``` ## 2. Cauchy-Affine Predictive Distribution Folder: ```text toy/cauchy-affine-response-kernel/ ``` Read second because it connects the HCGM spine to the current learnability seed: ```text factual predictors x^F -> Cauchy unit abduction candidate u + query x^Q + affine generator -> closed-form Cauchy predictive outcome distribution ``` What to notice: - Cauchy is learner uncertainty about the same actual token, not physical identity redraw. - Its parameters are location and scale; ordinary mean and variance do not exist. - Affine/linear response makes propagation analytic. - The toy supports predictive-distribution propagation, not full causal identification. Reader checkpoint: ```text Can you state what this toy proves as a predictive-distribution calculation, and what it does not prove about broad end-to-end learning? ``` ## 3. Simpson Paradox Folder: ```text toy/simpson-paradox/ ``` Read third because it shows how aggregate trends can reverse group-level mechanisms. What to notice: - The same observed \(X,Y\) trend can be misleading when latent group/unit mixture changes. - HCGM should ask which unit distribution is being used for the query. - The current figure and local robustness assets are teaching artifacts, not a full benchmark. Reader checkpoint: ```text Can you distinguish a population-level regression line from the unit or stratum mechanisms that would answer the target causal query? ``` ## 4. Berkson Paradox Folder: ```text toy/berkson-paradox/ ``` Read fourth because it isolates selection/collider distortion. What to notice: - Conditioning on selection can manufacture dependence. - HCGM should ask whether the observed sample represents the target unit distribution. - Selection robustness assets are local teaching/evaluation seeds, not a global HCGM benchmark. Reader checkpoint: ```text Can you explain why the selected population is not automatically the target counterfactual population? ``` ## 5. Surrogate-Marker Paradox Folder: ```text toy/surrogate-marker-paradox/ ``` Read fifth because it shows proxy optimization and path-level failure. What to notice: - A treatment can improve a surrogate marker while hurting the true outcome. - A model that predicts the proxy well can still fail the target causal query. - HCGM should keep outcome mechanism, proxy path, and target intervention separate. Reader checkpoint: ```text Can you explain why optimizing an observed proxy is not the same as improving the target outcome mechanism? ``` ## Minimal Completion Test A reader has completed the teaching path when they can answer: ```text What evidence updates U? What target condition enters f(x,E;u)? What unit distribution is being reduced over? What event noise is resampled? Which observed trend, selection rule, or proxy path could be misleading? ``` ## Boundary Do not expand this file into a full tutorial yet. Next grow should be one of: ```text add one diagram per checkpoint turn the five checkpoints into entry-site teaching cards turn the comparator contrasts into entry-site teaching cards promote one checkpoint into the regression paper as a motivation paragraph ``` For entry-site teaching cards, start from the teaching-card seed table in `TEACHING_CONTRASTS.md`. A card should not be treated as ready unless it has one comparator, one toy anchor, one HCGM first question, one misconception prevented, one evidence anchor, and one non-claim.