Re: defining averages over unknown PDF

*To*: mathgroup at smc.vnet.net*Subject*: [mg131948] Re: defining averages over unknown PDF*From*: Itai Seggev <itais at wolfram.com>*Date*: Wed, 6 Nov 2013 00:33:03 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net*References*: <20131105041659.58DD96A05@smc.vnet.net>

On Mon, Nov 04, 2013 at 11:16:59PM -0500, Sune wrote: > Dear all. > > I want to do some symbolic manipulations of an expression involving averages over a stochastic variable with an unknown density. Therefore, I figured I could define a function av, which would correspond to the average over this unknown parameter density function. > I did as follows: > av[y_ + z_, x_] := av[y, x] + av[z, x]? > av[c_ y_, x_] := c av[y, x] /; FreeQ[c, x] > av[c_, x_] := c /; FreeQ[c, x] > > So these are basic properties of the average over the distribution of X. Some things work okay, for example > In[52]:= av[Exp[-x y], x]? > Out[52]= av[E^(-x y), x] > and > In[79]:= D[av[-x y, x], x]? > Out[79]= -y > and > In[80]:= D[av[-x y, x], y]? > Out[80]= -av[x, x]. > > However, the most vital part for my calculations does not work: > In[81]:= D[av[Exp[-x y], x], y]? > Out[81]= -E^(-x y) x > > It should have been av[-Exp[-x y] x,x]. > > Any clues to what I'm doing wrong? I'm thinking that I need to specify some rules for differentiation, but I don't know how. But then I'm wondering how come it got the other expressions for differentiation right. Ahh, the subtle treacheries of partial differentiation. Note that by your definition, In[71]:= av[Exp[-x y] + h, x] - av[Exp[-x y], x] Out[71]= h So that In[72]:= Limit[(av[Exp[-x y] + h, x] - av[Exp[-x y], x])/h, h -> 0] Out[72]= 1 So both your "correct" and "incorrect" answers are consistent with the chain rule and and the above computation of partial derivatives. So why is D computing the partial derivative in such a stupid way? Well, it isn't, at least not directly. D correctly computes the partial derivative as f'[x] * Derivative[1, 0][av][f[x], x] + Derivative[0, 1][av][f[x],x] But now Derivative helpfully tries compute these partials using pure functions, and then your definitions kick in, giving 1 and 0 for the partials. In particular, your third definitions means av[#1,#2]& === #1, and you're doomed. So you want to abort the automatic differentiation rules with your own custom rule, which you can do with the following syntax: av /: D[av[f_, x_], y_] /; x =!= y := av[D[f, y], x] In[65]:= D[av[Exp[-x y], x], y] Out[65]= -av[E^(-x y) x, x] -- Itai Seggev Mathematica Algorithms R&D 217-398-0700

**References**:**defining averages over unknown PDF***From:*Sune <sunenj@gmail.com>