I am reading an article which is trying to justify the need for causal inference in their inferential framework. The thought experiment is as follows:

Suppose a statistician is asked to design a model for a simple time series $X_1,X_2,X_3,...$ and she decides to use a Bayesian method. Assume she collects a first observation $X_1 = x_1$. She computes the posterior probability density function (pdf) over the parameters $\theta$ of the model given the data using Bayes’ rule: $$p(\theta|X_1 = x_1) = \int\frac{p(X_1 = x_1|\theta)p(\theta)}{p(X_1 = x_1|\theta')p(\theta')}, $$

where $p(X_1 = x_1|θ)$ is the likelihood of $x_1$ given $\theta$ and p($\theta$) is the prior pdf of $\theta$. She can use the model to predict the next observation by drawing a sample $x_2$ from the predictive pdf: $$p(X_2 = x_2|X_1 = x_1) = \int p(X_2 = x_2|X_1 = x_1,\theta)p(\theta|X_1 = x_1)d\theta,$$

where $p(X_2 = x_2|X_1 = x_1,\theta)$ is the likelihood of $x_2$ given $x_1$ and $\theta$. Note that $x_2$ is not drawn from $p(X_2 = x_2|X_1 > = x_1, \theta)$. She understands that the nature of $x_2$ is very different from $x_1$: while $x_1$ is informative and does change the belief state of the Bayesian model, $x_2$ is non-informative and thus is a reflection of the model’s belief state. Hence, she would never use $x_2$ to further condition the Bayesian model. Mathematically, she seems to imply that: $$p(\theta|X_1 =x_1,X_2 =x_2)=p(\theta|X_1 =x_1)$$

However I hardly believe that what this poor statistician should imply is: $$p(\theta|X_1 =x_1,\text{do}(X_2 =x_2))=p(\theta|X_1 =x_1)$$ Where "do(or set)" here comes from Pearl's framework of causality which can be found here and here. Now am I right about this?

  • $\begingroup$ What does "do" mean in your final formula? $\endgroup$ – whuber Oct 13 '13 at 17:12
  • $\begingroup$ @whuber edited. $\endgroup$ – Cupitor Oct 13 '13 at 17:16
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    $\begingroup$ Thanks, but questions need to be understandable on their own, so if you can, please describe what this means rather than just providing links. Otherwise you may severely limit the potential audience for this question and reduce your chances of getting great answers. $\endgroup$ – whuber Oct 13 '13 at 17:19
  • $\begingroup$ Well I am gonna be honest about this. My understanding of causal framework is very primitive and naive and as stated in the paper I linked to is: "expressions of the form $P(Y = y|\text{set}(X = x))$ or $P(Y = y|\text{do}(X = x))$ to denote the probability (or frequency) that event $(Y = y)$ would occur if treatment condition $X = x$ were enforced uniformly over the population." You are right, but I thought new causal framework is a well known one among statisticians. What I am implying is the "statistician" above, should be a dumb to treat $X_2=x_2$ as an observational variable. $\endgroup$ – Cupitor Oct 13 '13 at 17:35

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