small fix for Ben

Signed-off-by: Nathaniel <[email protected]>

Author: NathanielF

docs/source/knowledgebase/structural_causal_models.ipynb

@@ -41,15 +41,15 @@
    "source": [
     "\n",
     "\n",
-    "When we ask \"What is the effect of a medical treatment?\" or \"Does quitting smoking cause weight gain?\" or \"Do job training programs increase earnings?\", we are not simply asking about the treatment itself. We are asking: What world are we operating in? This perspective is more easily seen if you imagine a causal analyst as a pet-shop owner introducing a new fish to one of their many acquariums. The new fish's survival and behavior depend less on its intrinsic properties than on how it fits within this complex, interconnected system of PH balances and predators. In which tank will the new fish thrive?  \n",
+    "When we ask \"What is the effect of a medical treatment?\" or \"Does quitting smoking cause weight gain?\" or \"Do job training programs increase earnings?\", we are not simply asking about the treatment itself. We are asking: What world are we operating in? This perspective is more easily seen if you imagine a causal analyst as a pet-shop owner introducing a new fish to one of their many acquariums. The new fish's survival and behavior depend less on its intrinsic properties than on how it fits within this complex, interconnected system of pH balances and predators. In which tank will the new fish thrive?  \n",
     "\n",
     "Different causal methods make different choices about how much of this system to model explicitly. Some methods succeed by not modeling the full system: instrumental variables isolate causal effects through credible exclusion restrictions; difference-in-differences leverages parallel trends; interrupted time-series assumes stationarity. These design-based approaches gain power by minimizing modeling assumptions about the data-generating process. See {cite:t}`pearl2000causality` or {cite:t}`angrist2009mostly` for more detailed distinctions. The unifying thread between these diverse methods is the idea of a causal model as a _probabilistic program_ : an inferential routine designed to explicitly yield insights into the effect of some intervention or treatment on the system of interest. Whether design based or model-based, causal inference methods assume a data generating process - the distinction between these methods is how explicitly the system is rendered.\n",
     "\n",
     "#### Modelling Worlds and Counterfactual Worlds\n",
     "\n",
     "Bayesian structural modeling attempts to parameterize the system itself. Where design-based methods answer \"what is the causal effect under these identification assumptions?\", structural models ask \"what is the most plausible data-generating process, and how do interventions propagate through it?\". In Bayesian structural causal inference the focus is slightly different in that we wish to model both the treatment and the environment i.e. the fish and the fishtank. The trade-off is transparency for complexity. You must specify more of the data-generating process, which creates more opportunities for model misspecification. But every assumption becomes an explicit, testable model component rather than an implicit background condition.\n",
     "\n",
-    "This is a two step move in the Bayesian paradigm. First we infer \"backwards\" what is the most plausible state of the world $w$ conditioned on the observable data. The \"world\" of the model is defined by: (1) a causal graph relating variables, (2) likelihood functions specifying how each variable depends on its causes, and (3) prior distributions over parameters. Optionally, this may include latent confounders, measurement models, and selection mechanisms—each adding structural detail but also complexity. With this world in place, we continue to assess the probabilistic predictive distribution of treatment and outcome at the plausible range of counterfactual worlds. \n",
+    "This is a two step move in the Bayesian paradigm. First we infer \"backwards\" what is the most plausible state of the world $w$ conditioned on the observable data `X, T, O`. The \"world\" of the model is defined by: (1) a causal graph relating variables, (2) likelihood functions specifying how each variable depends on its causes, and (3) prior distributions over parameters. Optionally, this may include latent confounders, measurement models, and selection mechanisms—each adding structural detail but also complexity. With this world $w = \\{ \\alpha, \\beta_{1}, \\beta_{2} ... \\}$ in place, we continue to assess the probabilistic predictive distribution of treatment and outcome at the plausible range of counterfactual worlds. \n",
     "\n",
     "![](../_static/forwards_backwards.png)\n",
     "\n",