add the do operator to make the intervention more clear

Author: drbenvincent

docs/source/knowledgebase/quasi_dags.ipynb

@@ -675,20 +675,20 @@
     "- $Y_1^{(1)}$ is the potential outcome if the unit receives treatment ($T=1$).\n",
     "- $Y_1^{(0)}$ is the potential outcome if the unit does not receive treatment ($T=0$).\n",
     "\n",
-    "For treated units ($T=1$):\n",
+    "For treated units:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
-    "Y_1^{(1)} = \\mathbb{E}[Y_1 \\,|\\, T=1, Y_0, \\mathbf{X}]  &= \\alpha + 1 \\tau \\times + \\gamma Y_0 + \\mathbf{X}\\\\\n",
+    "Y_1^{(1)} = \\mathbb{E}[Y_1 \\,|\\, \\text{do}(T=1), Y_0, \\mathbf{X}]  &= \\alpha + 1 \\tau \\times + \\gamma Y_0 + \\mathbf{X}\\\\\n",
     "                                                        &= \\alpha + \\tau + \\gamma Y_0 + \\mathbf{X}\\boldsymbol\\beta\n",
     "\\end{aligned}\n",
     "$$\n",
     "\n",
-    "For untreated units ($T=0$):\n",
+    "For untreated units:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
-    "Y_1^{(0)} = \\mathbb{E}[Y_1 \\,|\\, T=0, Y_0, \\mathbf{X}] &= \\alpha + 0 \\tau + \\gamma Y_0 + \\mathbf{X}\\boldsymbol\\beta\\\\\n",
+    "Y_1^{(0)} = \\mathbb{E}[Y_1 \\,|\\, \\text{do}(T=0), Y_0, \\mathbf{X}] &= \\alpha + 0 \\tau + \\gamma Y_0 + \\mathbf{X}\\boldsymbol\\beta\\\\\n",
     "                                                       &= \\alpha + \\gamma Y_0 + \\mathbf{X}\\boldsymbol\\beta\n",
     "\\end{aligned}\n",
     "$$\n",