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Re: Don't understand replacement rule for some functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89558] Re: Don't understand replacement rule for some functions
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 12 Jun 2008 06:30:53 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g2qhhr$8or$1@smc.vnet.net>

Mac wrote:

> Can somebody explain why the following replacement rule works
> 
> In[15]:= f[x] /. x -> y
> 
> Out[15]= f[y]
> 
> and this one complains
> 
> In[14]:= RandomReal[GammaDistribution[L, 1/L], 2] /. L -> 3
> 
> During evaluation of In[14]:= RandomReal::parpos: The parameter L in \
> position 1 of GammaDistribution[L,1/L] should be positive >>
> 
> Out[14]= {1.89311, 1.11748}
> 
> I find it quite elegant to use replacements for complicated functions
> but I'm never quite sure whether the replacement is actually carried
> out because of such behavior.

Before evaluating an expression, Mathematica does many things [1]. Among 
these things, it evaluates the arguments of the the expression *before* 
applying any transformation rules. Also, keep in mind that an expression 
is also a tree, so this process is applied recursively at each layer (or 
level) of the tree. Then, starting from the lowest level, the 
sub-expressions are evaluated in turn and any associated transformation 
rules at the same level is applied.

So, the evaluation of an expression such as f[x]/.x->y (assuming f, x, 
and y have no associated meaning) can be summarized as follows:

The expression f[x]/.x->y is split in two parts: f[x] and x->y.

Before applying the rule x->y, Mathematica evaluates the argument (i.e. 
x) of f[x]. Since x is an atomic expression with no value attached to 
it, it evaluates to itself, that is x.

So, now we are left with f[x] to which the rule x->y is going to be 
applied. Having done so, f[x] is transformed into f[y] and only now 
Mathematica is going to evaluate this expression f[y]. Since f has no 
definition attached to it, the result of the evaluation is f[y] itself.

Wow, I hope I have not confused you insofar!

Following the same procedure, this time with the second expression,

     RandomReal[GammaDistribution[L, 1/L], 2] /. L -> 3

we now understand that Mathematica attempts to evaluate first (there is 
a mechanism to control this, see below *) the arguments provided to 
RandomReal, that is 2, which evaluates to 2, and GammaDistribution[L, 
1/L] which evaluates to itself since GammaDistribution required numeric 
arguments (as well as RandomReal, thus the warning message).

Therefore, the result of this preprocessing, is the expression

     RandomReal[GammaDistribution[L, 1/L], 2]

to which Mathematica is going to apply the rule L->3 that was put on 
hold. Having applied the rule, the expression becomes

     RandomReal[GammaDistribution[3, 1/3], 2]

expression which is now perfectly valid and RandomReal is going to be 
evaluated, which yields the two random numbers you got after the warning 
message.

Consequently, as a rule of thumb, one should "attach" any transformation 
rules as close as possible of the relevant function or part of the 
expression. For instance,

In[1]:= RandomReal[GammaDistribution[L, 1/L] /. L -> 3, 2]

Out[1]= {0.382089, 0.471758}

Or, one can use the With[] construct as in

In[2]:= With[{L = 3}, RandomReal[GammaDistribution[L, 1/L], 2]]

Out[2]= {1.22402, 0.395845}

Well, I hope I have shed more light than obscurity on the process :-)]

Best regards,
- Jean-Marc

[1] "The Main Loop", 
http://reference.wolfram.com/mathematica/tutorial/TheMainLoop.html

* to see the attributes of a function, one can use Attributes[], and 
SetAttributes[] to set some specific attributes, such as HoldFirst that 
  delays the evaluation of the first argument until the execution of the 
body of the function.


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