<div leftalign>
** <- [[confounding-simpson|Confounding]] **
</div>

====== Probabilistic vs. Causal inference ======

This post revisits the [[confounding-simpson|confounding phenomenon]] under the 
angle of probabilistic reasoning with [[wp>conditional probabilities]] 
and [[wp>Bayes' theorem]].  This analysis reveals what
sets probabilistic and causal inference apart. It also
shows how probabiistic models and causal models
formulate assumptions of a fundamentally different nature.

**Tip**:
Feel free to 
[[confounding-bayes#Conclusion|jump to the conclusion]] 
if you find this discussion too technical.

===== Treatments and outcomes =====

Suppose for instance that we have access to a large trove of medical reports describing 
the outcomes of applying medical treatment to patients displaying certain symptoms.
In the following, we use variable $Y$ to denote observable outcomes such 
as "patient is considered cured after five days"; we use variable $X$ to denote
the treatment selected by a doctor; and we use variable $Z$ to represent the 
known contextual information. Although we refer to $Z$ as the symptoms variable for simplicity, 
it really represents all the observed and recorded context of the treatment decision:
patient symptoms, patient history, hospital history, place and time, or even 
environmental factors such as the pollution level.

//We merely want to know which treatment $X$ works best for symptoms $Z$.//

This question is natural but also ambiguous. Did we mean
  * //Which treatment $X$ has historically worked best for patients with specific symptoms $Z$//, or
  * //Which treatment $X$ would work best for patients with specific symptoms $Z$//?

Confounding exists because these two questions often have different answers.

==== Conditioning ====

To answer the first question, we merely need to go through our medical records
and count the observed outcomes $Y$ for each symptom-treatment pair $(Z,X)$.
If we assume that our records are independent samples from a joint probability 
distribution $P(Z,X,Y)$, this amounts to estimating the conditional 
probability distribution $P(Y|Z,X)$.

Since it is very unlikely that our trove of medical reports covers all possible combinations 
of symptoms $Z$ and treatments $X$ we may have no data to estimate $P(Y|Z,X)$ when $(Z,X)$ 
is one of these missing pairs. This is not a problem if our goal is merely to answer the 
first question because saying which treatments have historically worked best does
not require us to know anything about treatments we never tried. In contrast, the second 
question is more demanding because the best treatment for symptoms $Z$ could possibly 
be a treatment that was never tried for such symptoms. 
This difference is only the tip of a much larger iceberg. 

==== Interventions ====

Essentially the second question envisions an //intervention// to change the treatment 
selection policy. Instead of doing whatever the doctors were doing to select a 
treatment during the data collection period, we are considering an alternate treatment 
selection policy and we would like to know about its performance.

We can approach such an intervention by decomposing the joint distribution as 
\begin{equation}
   \label{eq:xyz}
    P(Z,X,Y) = P(Z) \, P(X|Z) \, P(Y|Z,X)~.
\end{equation}
This particular ordering of the variables is attractive because it
obeys the arrow of time: first come the symptoms with distribution $P(Z)$,
then comes the treatment selection with distribution $P(X|Z)$, 
and finally comes the treatment outcome with distribution $P(Y|X,Z)$.

<div centeralign center 67%>
{{ confounding-decomp-xyz.png?200 |Decomposition ZXY}}
// Graphical representation[(bibtex:Pearl1988)] of the decomposition \eqref{eq:xyz}.\\
Each node in this graph represents one variable and receives arrows from all 
variables that condition the corresponding term in the decomposition.//
</div>

We would like to know how our medical records would look like if
we were to follow an alternate policy to select a treatment $X$
given symptoms $Z$. This calls for replacing the term $P(X|Z)$ 
in the decomposition \eqref{eq:xyz} by a new conditional 
distribution $\textcolor{red}{P^*(X|Z)}$ that represents
the alternate treatment selection policy.
\begin{equation}
\label{eq:xyz:interv}
   \textcolor{red}{P^*(Z,X,Y)} = P(Z) \, \textcolor{red}{P^*(X|Z)} \, P(Y|Z,X)~.
\end{equation}

Does the new joint distribution $\textcolor{red}{P^*(Z,X,Y)}$
represents what one could observe if one were to implement the new treatment policy?  
Equivalently, do the conditional distributions $P(Z)$ and $P(Y|Z,X)$ remain 
//invariant// when one changes the treatment policy $P(X|Z)$.

<div center round box important 90%>
**Nontrivial assumption:** Changing the treatment selection policy in 
decomposition \eqref{eq:xyz} leaves the terms 
$P(Z)$ and $P(Y|Z,X)$ invariant. 
</div>

Such an assumption is not a part of our probabilistic model of the data,
namely the assumption that the medical records were independently sampled
from a joint distribution $P(Z,X,Y)$. To determine whether this assumption 
is correct or not, we must invoke pieces of knowledge that do not change
the probabilistic model itself. These pieces of knowledge are often
formulated in causal terms:


  * The distribution of symptoms $P(Z)$ could be changed by causal factors beyond our control such as the appearance of new viral strains. We can avoid such difficulties by reformulating our question with a different tense : //which treatment selection policy would have worked best if it had been applied to the cases described in our trove of medical records?// This is called a //counterfactual question// because it pertains to something that did not happen and cannot happen anymore. Since this question it applies to past cases, the unobserved environmental factors can be assumed to remain identical. Meanwhile, answering this counterfactual question still provides useful knowledge.
  * Since the symptoms are observed before the treatment selection, we can argue that the changing the treatment cannot change the symptoms. In the context of the counterfactual question, this means that their distribution $P(Z)$ is not affected by changes to the treatment selection policy $P(X|Z)$. 
  * The invariance of the conditional outcome distribution $P(Y|Z,X)$ is harder to justify. For instance, we may believe that the outcome $Y$ is a deterministic function of the observed symptoms $Z$, the selected treatments $X$, and, possibly, an independent roll of the dices. But what to conclude if we believe instead that the outcome could depend on symptoms that were not observed or not recorded in the medical records?


===== Unobserved variables ===== 

Probabilities are often used to represent processed that we cannot model completely.
For instance, what happens when we roll the dices might depend on their initial configuration,
the speed of our motion, the exact moment we release the dices, the temperature of the air, etc.
Since we do not know these things, we simply say that each face of a dice is shown with equal chance.

We shall now change our model to include a new variable $U$ that represents additional
information that was not recorded in the medical reports and therefore is no longer accessible. 
The exact nature of the information $U$ is not important at this point, except for the 
fact that it may impact which treatment was applied as well as its outcome. 
Such a variable $U$ is often called an //unobserved variable// because its value
is unknown at the time of the analysis of the data. 

==== Interventions in the presence of unobserved variables ====

The canonical decomposition then becomes
\begin{equation}
\label{eq:xyzu}
    P(Z,U,X,Y) = P(Z) \, P(U|Z) \, P(X|Z,U) \, P(Y|X,Z,U) ~.
\end{equation}
<div centeralign center 80%>
{{ confounding-decomp-xyzu.png?280 |Decomposition ZXY}}
//Canonical decomposition with an unobserved variable $U$.//
</div>
We can then write the observed joint distribution $P(Z,X,Y)$ as
\begin{align*}
     P(Z,X,Y) &= \sum_U P(Z,U,X,Y) \\
              &= P(Z) \, \sum_U P(U|Z) \, P(X|U,Z) \, P(Y|X,U,Z)~.
\end{align*}

The term $P(X|Z,U)$ that represents which treatments $X$ were
applied now depends on both the symptoms $Z$ and the unobserved variable $U$. 
Proceeding as in the previous section, we replace this term
by the conditional distribution $\textcolor{red}{P^*(X|Z)}$ that
represents an alternate treatment policy. Note that the alternate 
policy cannot depend on $U$ because we do not know $U$.
This gives the following expression for the alternate joint distribution
\begin{equation}
\label{eq:xyzu:interv}
   \textcolor{red}{P^*(Z,X,Y)} = P(Z) \, \textcolor{red}{P^*(X|Z)} ~
      \textcolor{blue}{\underbrace{\sum_U P(U|Z) \, P(Y|Z,X,U)}_{P^*(Y|Z,X)} }
\end{equation}

<div center round box important 90%>
** New assumption: **
We now assume here that changing the treatment policy leaves
the terms $P(Z)$, $P(U|Z)$, and $P(Y|Z,X,U)$ invariant.
</div>

The nature of this new assumption depends of course on the assumed
nature of the unobserved variable $U$. For instance, if the variable
$U$ is constant, our new assumption is equivalent to our previous
assumption, namely the invariance of $P(Z)$ and $P(Y|Z,X)$.
We now would like to know whether there are choices of $U$
that make this assumption different and lead to different
conclusions.

This boils down to determining whether the blue term in \eqref{eq:xyzu:interv}
\[
   \textcolor{blue}{P^*(Y|Z,X) = \sum_U P(U|Z) P(Y|Z,X,U) } ~,
\]
is equal to
\[
    P(Y|Z,X) = \sum_U P(Y,U|Z,X) = \sum_U P(U|Z,X) P(Y|Z,X,U) 
\]
If these terms were equal, expressions \eqref{eq:xyzu:interv} and \eqref{eq:xyz:interv}
would be identical and therefore would lead to the same conclusions regardless of the nature
of the unknown information $U$. Unfortunately the first expression has $P(U|Z)$
where the second expression has $P(U|Z,X)$. 

These two conditional distribution are nevertheless equal in two important cases:

  * when the unobserved variable $U$ has no impact on the outcomes,that is, when $P(Y|Z,X,U)=P(Y|Z,X)$, and
  * when the unobserved variable $U$ had no impact on the selected treatments, that is, when $P(X|Z,U)=P(X|Z)$.


<div indent><typo fs:smaller; fc:#777;>**Proof**: 
In the first case, we can indeed write
\[ 
   P^*(Y|Z,X) = \sum_U P(U|Z) P(Y|Z,X,U) 
     = \big( \sum_U P(U|Z) \big) P(Y|Z,X) = P(Y|Z,X)~.
\]
For the second case, decomposing $P(X,U|Z)$ in two ways,
\[  P(X,U|Z) = P(X|Z) P(U|Z,X) = P(U|Z) P(X|Z,U)~, \]
shows that $P(X|Z,U){=}P(X|Z) \Longleftrightarrow P(U|Z){=}P(U|Z,X)$.
Therefore $P^*(Y|Z,X){=}P(Y|Z,X)$. $~\blacksquare$
</typo></div>

<div clear/>


However, when the unobserved variable $U$ impacts both the treatments and the outcomes, the two conditional distributions $\textcolor{blue}{P^*(Y|Z,X)}$ and $P(Y|Z,X)$ are not equal. 
Since equations \eqref{eq:xyz:interv} and \eqref{eq:xyzu:interv} are different, 
it is very likely that these two approaches lead to  different 
joint distributions $\textcolor{red}{P^*(Z,X,Y)}$ and therefore 
different conclusions about which treatment works best for which symptom.
To make things worse, we cannot in general estimate $\textcolor{blue}{P^*(Y|Z,X)}$ 
as defined in equation \eqref{eq:xyzu:interv} because it depends on the unobserved 
variable $U$ which cannot be found in our medical records.
 

===== Conclusion =====

This example illustrates a fundamental difference between probabilistic and causal inference.

<div center round box 90%>
  * **Probabilistic inference**:  Knowing the joint distribution of the observed variables is enough to compute all the conditional distributions and therefore __sufficient__ to answer any probabilistic inference question pertaining to the observed variables.
  * **Causal inference**: Knowing the joint distribution of the observed variables is __generally not sufficient__ to answer a causal inference question pertaining to the observed variables: the outcome of interventions on the observed variables potentially __depends on unobserved parts of the system__.
</div>

This also implies that probabilistic and causal modeling are fundamentally different

<div center round box 90%>
  * **Probabilistic models** make it __easier__ to estimate the joint distribution of the observed variable. Making conditional independence assumptions or making parametric assumptions helps us to produce an estimate using only a limited number of examples.
  * **Causal models** potentially make it __possible__ to estimate the outcome of interventions. We simply __cannot__ give an answer without making __invariance assumptions__ or making __independence assumptions__ with respect to unobserved quantities.
</div>


<div rightalign>
** [[causal-models|Many kinds of causal models]] -> **
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