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Re: defining averages over unknown PDF

  • To: mathgroup at smc.vnet.net
  • Subject: [mg131999] Re: defining averages over unknown PDF
  • From: Itai Seggev <itais at wolfram.com>
  • Date: Thu, 14 Nov 2013 02:10:44 -0500 (EST)
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  • References: <20131105041659.58DD96A05@smc.vnet.net>

Typo in my previous email.  There should have been no undscore, since f is the
function we're binding to:

f /: Derivative[n_][f] := (-1)^n f

Also, it would have been better to write this as 

f /: Derivative[n_][f] := (-1)^n f[#]&

or we'd end up with the compond head -f acting on its arguments for odd orders.

As for formulating things in terms of Derivative, I don't think that will solve
your problem.  Because if you have av[f[x, y],x], then the chain rule will put
df on the outside of av, which may not be what you want.  It is rather unclear
how you could write the rules in terms of Derivative, but if you can that would
be the right solution.

On Tue, Nov 12, 2013 at 10:02:35AM +0100, Sune Jespersen wrote:
> Thanks a lot for taking the time to look into this!
> 
> From your description, it sounds a little like I could just formulate the
> rule in terms of Derivative instead of D? Would that solve it?
> 
> I'm relatively new to Mathematica, but I find it surprising that with the
> remarkable power of Mathematica, it is not possible to implement a
> relatively simple thing like this.
> 
> Sune
> 
> -----Original Message-----
> From: Itai Seggev [mailto:itais at wolfram.com] 
> Sent: 12. november 2013 06:04
> To: Sune Jespersen
> Cc: mathgroup at smc.vnet.net
> Subject: Re: Re: defining averages over unknown PDF
> 
> I've done some digging into this.  The normal way you define custom
> differentiation rules is by define UpValues for it and Derivative, like
> 
> f /: Derivative[n_][f_] := (-1)^n f  (*f is secrelt Exp[-x]*)
> 
> This way, all differentiation functions (of which there are D, Dt,
> Derivative, and several more internal and user-visible) share equally in the
> rule.  When you write your rule in terms of D and differentation av[...], D
> starts evaluating, the rule fires, and everything is fine.  When av is
> inside another function, the chain rule code fires and says that
> 
> Derivative[1][f[av]] := Derivative[1][f][av][#] * Derivative[1][av][#]&
> 
> Then this code looks up any rules for Derivative[1][av], finds none, and
> procedes to use automatic differentiation, getting you back to your initial
> state.
> 
> I'm afraid I don't have a really great solution for you.  D is really
> intended to differentiate functions, but av is an operator in disguies which
> makes this quite difficult to work with.  Getting a good operator calculus
> into the system is a problem I and others have thought about, but it is a
> long-term project which will not be solved in the short term. 
> 
> Sorry!
> 
> On Wed, Nov 06, 2013 at 12:33:43AM -0500, Sune Jespersen wrote:
> > Hi Itai
> > 	
> > Thanks for your reply. Yes, you are of course right and I realized the 
> > same thing shortly after my post. In fact I implemented a solution 
> > quite similar to yours, In[1]:= av /: D[av[f___], x_] := av[D[f, x]] 
> > In[2]:= av[y_ + z_] := av[y] + av[z] In[3]:= av[c_ y_] := c av[y] /; 
> > FreeQ[c, x] In[4]:= av[c_] := c /; FreeQ[c, x] In[5]:= D[av[x y], x] 
> > Out[5]= y In[6]:= D[av[Exp[-x y]], x] Out[6]= -y av[E^(-x y)] But it 
> > seems that it still has the problem when it needs to apply the chain rule,
> i.e.
> > In[9]:= D[Log[av[Exp[-b x]]], b]
> > Out[9]= -((E^(-b x) x)/av[E^(-b x)])
> > instead of
> > -(av[(E^(-b x) x)]/av[E^(-b x)])
> > This seems a bit strange to me, because somehow it must reach a point
> where it needs to evaluate a derivative, where my rule applies. Perhaps you
> can offer some insight on this?
> > 
> > On 5 Nov, 2013, at 18:19 , Itai Seggev <itais at wolfram.com> wrote:
> > 
> > > 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
> > 
> > 
> --
> Itai Seggev
> Mathematica Algorithms R&D
> 217-398-0700 
> 
--
Itai Seggev
Mathematica Algorithms R&D
217-398-0700 



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