docs: improve narrative flow, standardize math notation, and fill gaps

- Causal Inference: Clarify bare demonstrative pronouns, standardize formal
  definitions to use \eqdef, add toy numerical examples for potential outcomes
  and population estimands, detail step-by-step derivations for RCT exchangeability
  and G-computation linear model equivalence, update slidebreaks, and wrap IPW table in div.
  Standardize vector notation to use \vL and \vl.
- Predictor Selection: Fix div heading levels from ### to #### to prevent section
  numbering conflicts, resolve bare demonstratives, define VIF with \eqdef, and add
  numerical VIF example.
- Probability: Standardize theorem/definition div headings to level 4 (####), replace
  bare --- slidebreaks before theorem divs with {{< slidebreak >}}, and add step-by-step
  parenthetical justifications in proof aligned blocks.

Co-authored-by: dem-extra1 <112029334+dem-extra1@users.noreply.github.com>

Author: google-labs-jules[bot]

_subfiles/causal-inference/_sec_regression_adjustment.qmd

@@ -57,7 +57,8 @@ the average causal effect of $A$ on $Y$:
 $$
 \ba
 \text{ATE}
-&= \E{Y(1)} - \E{Y(0)} && \text{(definition of ATE)}
+&\eqdef \E{Y(1) - Y(0)} && \text{(definition of ATE)}
+\\&= \E{Y(1)} - \E{Y(0)} && \text{(linearity of expectation)}
 \\&= \E{\E{Y(1) \mid \vL}} - \E{\E{Y(0) \mid \vL}} && \text{(law of iterated expectations)}
 \\&= \E{\E{Y \mid A = 1, \vL}} - \E{\E{Y \mid A = 0, \vL}} && \text{(consistency and conditional exchangeability)}
 \\&= \E{\beta_0 + \beta_A (1) + \tp{\vbeta_L} \vL} - \E{\beta_0 + \beta_A (0) + \tp{\vbeta_L} \vL} && \text{(linear outcome model specification)}

chapters/probability.qmd

@@ -1039,8 +1039,11 @@ under the conditional distribution given $Z=z$:
 $$
 \ba
 \E{\E{Y \mid X,Z} \mid Z=z}
-&= \sum_{x \in \rangef{X}} \E{Y \mid X=x,Z=z} \cd \P(X=x \mid Z=z) && \text{(expectation under conditional distribution given } Z=z \text{)}
-\\&= \E{Y \mid Z=z} && \text{(law of total probability given } Z=z \text{)}
+&= \sum_{x \in \rangef{X}} \E{Y \mid X=x,Z=z} \cd \P(X=x \mid Z=z) && \text{(definition of expectation given } Z=z \text{)}
+\\&= \sum_{x \in \rangef{X}} \paren{\sum_{y \in \rangef{Y}} y \cd \P(Y=y \mid X=x,Z=z)} \cd \P(X=x \mid Z=z) && \text{(definition of conditional expectation } \E{Y \mid X=x,Z=z} \text{)}
+\\&= \sum_{y \in \rangef{Y}} y \cd \sum_{x \in \rangef{X}} \P(Y=y \mid X=x,Z=z) \cd \P(X=x \mid Z=z) && \text{(exchange order of summation)}
+\\&= \sum_{y \in \rangef{Y}} y \cd \P(Y=y \mid Z=z) && \text{(law of total probability given } Z=z \text{)}
+\\&= \E{Y \mid Z=z} && \text{(definition of conditional expectation given } Z=z \text{)}
 \ea
 $$