correct the discussion about backdoor path 3 and U_1

Author: drbenvincent

docs/source/knowledgebase/quasi_dags.ipynb

@@ -643,12 +643,12 @@
     "\n",
     "1. $T \\leftarrow Y_0 \\rightarrow Y_1$: this can be closed by observing $Y_0$. \n",
     "2. $T \\leftarrow X \\rightarrow Y_1$: this can be closed by observing the covariates $\\mathbf{X}$.\n",
-    "3. $T \\leftarrow Y_0 \\leftarrow U_1 \\to Y_1$: this is a real danger to unbiased estimation.\n",
-    "4. $T \\leftarrow U_2 \\rightarrow Y_1$: this is another potential danger to unbiased estimation.\n",
+    "3. $T \\leftarrow Y_0 \\leftarrow U_1 \\to Y_1$: this can be closed by observing $Y_0$.\n",
+    "4. $T \\leftarrow U_2 \\rightarrow Y_1$: this is danger to unbiased estimation.\n",
     "\n",
     "By conditioning on both $Y_0$ and $X$ we intercept backdoor paths 1 and 2, satisfying Pearl's back-door criterion. The treatment coefficient in the ANCOVA formula therefore identifies the causal effect of interest under the model's other assumptions (no measurement error, correct functional form, etc.).\n",
     "\n",
-    "If we can discount $U_1$ and $U_2$ then we can estimate the treatment effect $T \\to Y_1$ by running the linear regression:\n",
+    "If we can discount $U_2$ then we can estimate the treatment effect $T \\to Y_1$ by running the linear regression:\n",
     "\n",
     "$$\n",
     "Y_1 = \\alpha + \\tau\\,T + \\gamma\\,Y_0 + \\mathbf{X}\\boldsymbol\\beta + \\varepsilon ,\n",
@@ -708,7 +708,7 @@
     "If $\\tau$ was a scaler, then we simply have $\\text{ATE} = \\tau$. If $\\tau$ is instead a posterior distribution, then we can think of the posterior over $\\tau$ as the posterior over the ATE. In this way, the treatment coefficient $\\tau$ captures the ATE in the regression model.\n",
     ":::\n",
     "\n",
-    "However, if there are unobserved confounders $U_1$ or $U_2$, then these will pose real threats to unbiased estimation of the treatment effect $T \\to Y_1$."
+    "However, if there is an unobserved confounder $U_2$, then this will pose a real threat to unbiased estimation of the treatment effect $T \\to Y_1$."
    ]
   },
   {