{"id":34,"date":"2007-02-27T16:08:01","date_gmt":"2007-02-28T00:08:01","guid":{"rendered":"http:\/\/www.mii.ucla.edu\/causality\/?p=43"},"modified":"2007-02-27T16:08:01","modified_gmt":"2007-02-28T00:08:01","slug":"counterfactuals-in-linear-systems-2","status":"publish","type":"post","link":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/2007\/02\/27\/counterfactuals-in-linear-systems-2\/","title":{"rendered":"Counterfactuals in linear systems"},"content":{"rendered":"

What do we know about counterfactuals in linear models?<\/strong><\/font><\/p>\n

Here is a neat result concerning the testability of counterfactuals in linear systems.<\/strong>
We know that counterfactual queries of the form P<\/em>(Yx<\/sub><\/em>=y<\/em>|e<\/em>) may or may not be empirically identifiable, even in experimental studies. For example, the probability of causation, P<\/em>(Yx<\/sub><\/em>=y<\/em>|x',y'<\/em>) is in general not identifiable from experimental data (Causality<\/em>, p. 290, Corollary 9.2.12) when X<\/em> and Y<\/em> are binary.1<\/sup> (Footnote-1: A complete graphical criterion for distinguishing testable from nontestable counterfactuals is given in Shpitser and Pearl (2007, upcoming)).<\/p>\n

This note shows that things are much friendlier in linear analysis:<\/p>\n

Claim A. Any counterfactual query of the form E<\/em>(Yx<\/sub><\/em> |e<\/em>) is empirically identifiable in linear causal models, with e<\/em> an arbitrary evidence.<\/p>\n

Claim B. E<\/em>(Yx<\/sub><\/em>|e<\/em>) is given by <\/p>\n

E<\/em>(Yx<\/sub><\/em>|e<\/em>) = E<\/em>(Y<\/em>|e<\/em>) + T<\/em> [x<\/em> – E<\/em>(X<\/em>|e<\/em>)]      (1)<\/p>\n

where T<\/em> is the total effect coefficient of X<\/em> on Y<\/em>, i.e.,<\/p>\n

T<\/em> = d E<\/em>[Yx<\/sub><\/em>]\/dx<\/em> = E<\/em>(Y<\/em>|do<\/em>(x+1<\/em>)) – E<\/em>(Y<\/em>|do<\/em>(x<\/em>))      (2)<\/p>\n

Thus, whenever the causal effect T<\/em> is identified, E<\/em>(Yx<\/sub><\/em>|e<\/em>) = is identified as well.<\/p>\n


Claim A is not surprising. It has been established in generality by Balke and Pearl (1994b) where expressions involving the covariance matrix were used for the various terms in (1).<\/p>\n

Claim B offers an intuitively compelling interpretation of (1) that reads as follows: Given evidence e<\/em>, to calculate E<\/em>(Yx<\/sub><\/em> |e<\/em>), (i.e., the expectation of Y<\/em> under the hypothetical assumption that X<\/em> were x<\/em>, rather than its current value), first calculate the best estimate of Y<\/em> conditioned on the evidence e<\/em>, E<\/em>(Y<\/em>|e<\/em>), then add to it whatever change is expected in Y<\/em> when X<\/em> undergoes a forced increase from its current best estimate, E<\/em>(X<\/em>|e<\/em>), to its hypothetical value X=x<\/em>. That last addition is none other but the effect coefficient T<\/em>, times the expected change in X<\/em>, i.e., T<\/em>[x<\/em> – E<\/em>(X<\/em>|e<\/em>)]<\/p>\n

Note: Eq. (1) can also be written in do<\/em>(x<\/em>) notation as<\/p>\n

E<\/em>(Yx<\/sub><\/em>|e<\/em>) = E<\/em>(Y<\/em>|e<\/em>) + E<\/em>(Y<\/em>|do<\/em>(x<\/em>)) – E<\/em>[Y<\/em>|do<\/em>(X<\/em>=E<\/em>(X<\/em>|e<\/em>))]      (1') <\/p>\n

Proof:<\/strong>
(with help from Ilya Shpitser)<\/p>\n

Assume, without loss of generality, that we are dealing with a zero-mean model. Since the model is linear, we can write the relation between X<\/em> and Y<\/em> as:<\/p>\n

Y = TX + I + U<\/em>      (3)<\/p>\n

where T<\/em> is the total effect of X<\/em> on Y<\/em>, given in (2), I<\/em> represents terms containing other variables in the model, nondescendants of X<\/em>, and U<\/em> representing exogenous variables.<\/p>\n

It is always possible to bring the function determining Y<\/em> into the form (3) by recursively substituting the functions for each rhs variable that has X<\/em> as an ancestor, and grouping all the X<\/em> terms together to form TX<\/em>. Clearly, T<\/em> is the Wright-rule sum of the path costs originating from X<\/em> and ending in Y<\/em> (Wright, 1921).<\/p>\n

From (3) we can write:<\/p>\n

Yx<\/sub> = Tx + I + U<\/em>      (4)<\/p>\n

since I<\/em> and U<\/em> are not affected by hypothetical change from X=x<\/em> and, moreover,<\/p>\n

E<\/em>(Yx<\/sub><\/em>|e) = Tx + E<\/em>(I+U<\/em>|e<\/em>)      (5)<\/p>\n

since x<\/em> is a constant.<\/p>\n

The last term in (5) can be evaluated by taking expectations on both sides of (3), giving:<\/p>\n

E<\/em>(I+U<\/em>|x<\/em>) = EY<\/em>|e<\/em>) – TE<\/em>(X<\/em>|e<\/em>)      (6)<\/p>\n

and, substituted into (5), yields E<\/em>(Yx<\/sub><\/em>|e<\/em>) = Tx + E<\/em>(Y<\/em>|e<\/em>) – E<\/em>(X<\/em>|e<\/em>)      (7)
and proves our target formula (1).
——————– QED<\/p>\n

Some Familiar Problems Cast in Linear Outfits<\/strong>
Three Special cases of e<\/em> are worth noting:
Example-1. e<\/em>: X =x', Y = y'<\/em>
(The linear equivalent of the probability of causation) From (1) we obtain directly<\/p>\n

E<\/em>(Yx<\/sub><\/em>|Y=y', X=x'<\/em>) = y' + T<\/em> (x – x'<\/em>) <\/p>\n

This is intuitively compelling. The hypothetical expectation of Y<\/em> is simply the observed value of Y, y'<\/em>, plus the anticipated change in Y<\/em> due to the change x-x'<\/em> in X<\/em>.<\/p>\n

Example-2. e<\/em>: X = x'<\/em> (effect of treatment on treated)<\/p>\n

E<\/em>(Yx<\/sub><\/em>|X=x'<\/em>) = E<\/em>(Y<\/em>|x'<\/em>) + T<\/em> (x<\/em> – x'<\/em>)
            = rx' + T <\/em>(x – x'<\/em>)
                                  = rx' + E<\/em>(Y<\/em>|do<\/em>(x<\/em>)) – E<\/em>(Y<\/em>|do<\/em>(x'<\/em>)) where r<\/em> is the regression coefficient of Y<\/em> on X<\/em>. <\/p>\n

Example-3. e<\/em>; Y = y'<\/em>
(Gee, my temperature is Y=y'<\/em>, what if I had taken x<\/em> tablets of aspirin. How many did you take? Don't remember.)<\/p>\n

E<\/em>(Yx<\/sub> <\/em>|Y=y'<\/em>) = y' + T <\/em>[x – E<\/em>(X<\/em>|y'<\/em>)]
                                      = y' + E<\/em>(Y<\/em>|do<\/em>(x<\/em>)) – E<\/em>[Y<\/em>|do<\/em>(X=r'y'<\/em>)]<\/p>\n

where r'<\/em> is the regression coefficient of X<\/em> on Y<\/em>.<\/p>\n

Example-4. Let us consider the non-recursive, supply-demand model of page 215 in Causality<\/em> (2000). Eqs. (7.9)-(7.10) read:<\/p>\n

q = b1<\/sub>p + d1<\/sub>i +u1<\/sub><\/em>
p = b2<\/sub>q + d2<\/sub>w +u2<\/sub><\/em><\/p>\n

Our counterfactual problem (page 216) reads: Given that the current price is P=p0<\/sub><\/em>, what would be the expected value of the demand Q<\/em> if we were to control the price at P = p1<\/sub><\/em>? Making the correspondence P = X, Q = Y, e =<\/em> {P=p0<\/sub>, i, w<\/em>}, we see that this problem is identical to Example 2 above (effect of treatment on the treated), subject to conditioning on i<\/em> and w<\/em>. Hence, since T = b1<\/sub><\/em>, we can immediately write<\/p>\n

E<\/em>(Qp1<\/sub><\/sub><\/em> | p0<\/sub>, i, w<\/em>) = E<\/em>(Y<\/em>|p0<\/sub>,i,w<\/em>) + b1<\/sub><\/em>(p1<\/sub> – p0<\/sub><\/em>)
        &
\nnbsp;                                  = rp<\/sub> p0<\/sub> + ri<\/sub> i + rw<\/sub> w + b1<\/sub><\/em>(p1<\/sub>-p0<\/sub><\/em>)     (8)<\/p>\n

where rp<\/sub>, ri<\/sub><\/em> and rw<\/sub><\/em> are the coefficient of P, i<\/em> and w<\/em>, respectively, in the regression of Q<\/em> on P, i<\/em> and w<\/em>.<\/p>\n

Eq. (8) replaces Eq. (7.17) on page (217). Note that the parameters of the price equation<\/p>\n

p = b2<\/sub>q + d2<\/sub>w +u2<\/sub><\/em><\/p>\n

only enter (8) via the regression coefficients. Thus, they need not be calculated explicitly in case the are estimated directly by least square.<\/p>\n

Remark 1:<\/strong>
Example 1 is not really surprising; we know that the probability of causation is empirically identifiable under the assumption of monotonicity (Causality<\/em>, p. 293). But examples 2 and 3 trigger the following conjecture:<\/p>\n

Conjecture<\/strong>
Any counterfactual query of the form P<\/em>(Yx<\/sub><\/em> |e<\/em>) is empirically identifiable when Y<\/em> is monotonic relative to X<\/em>.<\/p>\n

It is good to end on a challenging note.<\/p>\n

Best wishes,
========Judea Pearl <\/font><\/p>\n","protected":false},"excerpt":{"rendered":"

What do we know about counterfactuals in linear models? Here is a neat result concerning the testability of counterfactuals in linear systems. We know that counterfactual queries of the form P(Yx=y|e) may or may not be empirically identifiable, even in experimental studies. For example, the probability of causation, P(Yx=y|x',y') is in general not identifiable from […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,22],"tags":[],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-counterfactual","category-linear-systems"],"_links":{"self":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/34","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=34"}],"version-history":[{"count":0,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/34\/revisions"}],"wp:attachment":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=34"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=34"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=34"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}