This classic work of economic humor/broadside against econometric inference/clever application by Heckman has finally been published in Economic Inquiry after floating around the past three decades. Here’s the setup: Let the population density of prayer be represented by f(x), scaled to [0,1] and let Y be an unobserved [0,1] variable representing God’s attitude toward mankind. Assume (“accept of faith”, as Heckman has it) that the conditional density g(X|Y)=a(y)exp(xy) where a is any continuous, positive, differentiable function. A proof by the statistician Singh shows that we can then infer that E(y|X=x)=f'(x)/f(x); that is, even without knowing the function a(y) or knowing any value of y, we can learn God’s attitude toward man, conditional on prayer! Given the pattern of US prayer, a little prayer turns out to make God angrier, whereas He looks favorably on a lot.
The (very short) paper concludes with a response from the sociologist/Catholic priest Father Greeley.