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 macros to \vL, \vl, and estimator indirection macros \Est and \emu.
- 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. Remove abbreviation in proof text.

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

Author: google-labs-jules[bot]

_subfiles/causal-inference/_sec_propensity_scores.qmd

@@ -72,7 +72,7 @@ $$w_i \eqdef \frac{A_i}{\hat{e}(\vL_i)} +
 \frac{1 - A_i}{1 - \hat{e}(\vL_i)}$$
 
 The IPW estimator of $\E{Y(a)}$ is:
-$$\widehat{\E{Y(a)}}_{\text{IPW}} \eqdef
+$$\Est{\E{Y(a)}}_{\text{IPW}} \eqdef
 \frac{\sum_{i: A_i = a} w_i Y_i}{\sum_{i: A_i = a} w_i}$$
 
 Reweighting by $w_i$ creates a **pseudo-population**

_subfiles/causal-inference/_sec_regression_adjustment.qmd

@@ -13,17 +13,17 @@ The **G-computation** estimator
 estimates the ATE by:
 
 1. Fitting a regression model:
-   $\hat{\mu}(a, \vl) \eqdef \widehat{\E{Y \mid A = a, \vL = \vl}}$
+   $\emu(a, \vl) \eqdef \Est{\E{Y \mid A = a, \vL = \vl}}$
 
 2. Predicting the potential outcome mean for each individual
    under both treatment levels $a = 1$ and $a = 0$:
-   $$\hat{Y}_i(a) \eqdef \hat{\mu}(a, \vL_i)$$
+   $$\hat{Y}_i(a) \eqdef \emu(a, \vL_i)$$
 
 3. Averaging over the study population:
-   $$\widehat{\E{Y(a)}} \eqdef \frac{1}{n} \sum_{i=1}^n \hat{Y}_i(a)$$
+   $$\Est{\E{Y(a)}} \eqdef \frac{1}{n} \sum_{i=1}^n \hat{Y}_i(a)$$
 
 4. Estimating the ATE as the contrast:
-   $$\widehat{\text{ATE}} \eqdef \widehat{\E{Y(1)}} - \widehat{\E{Y(0)}}$$
+   $$\Est{\text{ATE}} \eqdef \Est{\E{Y(1)}} - \Est{\E{Y(0)}}$$
 
 ::: notes