The idle ramblings of a Jack of some trades, Master of none

If an action A results in an effect E, but the effect E is not measurable (except, possibly, by emotional response), what can we say about it? In other words, can we make some mathematical statements describing E? To turn the question around, can this unobservable variable say anything about the efficacy of the action A?

This may seem like gobbledygook, but it is a statistical problem that was addressed by R.S. Singh in a paper 1 published thirty-one years ago. It may be of interest in, say, sociology, where a combination of causes results in an indiscriminate response, and one wants to determine if this response is real, and if so, what kind of response it is.

Recently, James Heckman 2 used this technique to ascertain God's attitude (the effect E) towards prayers (the action A).

After you pick yourselves off the floor and close your mouths, please pay attention here. From time immemorial, men and women have prayed to the divinity, asking for mercy, solace, money, help. What has the divinity's response been to these myriads of requests? Heckman shows how one might begin to answer this deep imponderable.

Assume that God's attitude is an (unknowable, unmeasurable) variable Y. Let X be the variable (ranging between 0 and 1) that describes the intensity of prayer. X, we can safely say, is a distribution with two humps: many people don't pray at all, and many pray a lot. He obtains the hard data on the distribution of faith and intensity of prayers from the National Opinion Research Center. Let the density of X be f(X). Let us assume (on faith - and what is life without some faith?) that the conditional density of X given Y is of an exponential form:

g(X\mid Y)=a(Y)e^{^{XY}}

where a(Y) is an unknown function with certain properties that needn't concern us at the moment.

Then we can compute the conditional expectation of Y given X by means of Singh's result:

E(Y\mid X=x)=\frac{f'(x)}{f(x)}

Using data from the NORC, Heckman was able to compute a table of expected values of God's attitude given the intensity of prayer. He concluded that a little prayer did no good and could make things much worse. A lot of prayer helped a lot.

Now if only we could believe our distributional assumptions. We'd all be better off praying constantly...

(See also Pascal and Atheism.)


Singh, R.S. (1977). Applications of estimators of a density and its derivatives to certain statistical problems. Journal of the Royal Statistical Society: Series B (Methodological) 39 (3), pp 257-363.

Heckman, J. (2008). The Effect of Prayer on God's Attitude Toward Mankind, IZA Discussion Series, No 3636.


Shefaly said...


Interesting post and idea. A particular approach to qualitative research (that I used) involves approximating a qualitative factor to a quantitative one through greater and greater granularity and detail. Hopefully this is good fodder for when we manage to have coffee. :-)

Fëanor said...

Shefaly: I suspect that that coffee can be described by the (unmeasurable) variable Y :-)

V Ramesh said...


As I said in one of my emails to you, there is a relation to prayers and good deeds (not of bad deeds) happenings. The happening of good deeds would boost praying. so if we map the happenings of good deed to god's attitude, still this looks like a open loop system to me. I feel a closed loop system would appeal better


Fëanor said...

Ramesh: Clearly you've forgotten the Gods of Hack who, if you prayed too much to them, would curse you, or, if they were totally pissed off, kill you dead! There's nothing in one's experience to show that only good consequences arise from prayer. After all, when things go well, it's God's grace, but when they go badly, who are we to question His wisdom? So I'm not sure I agree with your open or closed system interpretation.

efrique said...

The link to the Heckman paper is wrong (you link to paper 3856, instead of 3636).

Fëanor said...

efrique: You are right! I've fixed the link.

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