[Another response in this thread from David Kohel (who maybe should be
posting on list)]
Hi William and Paul,
Actually, I correct myself -- the average should be over the values
of the function, weighted by the probabilities. The domain of the
function (the keys) can be in any set (e.g. "A","B","C"), so the
current behavior is correct.
In the defintition of the probability space itself:
sage: ps = DiscreteProbabilitySpace([1,2,3],{1:1/3,2:1/3,3:1/3})
Consider replacing the domain of ps with such a set. Then it
should be clear that you can't average over "A", "B", and "C".
S = ["A","B","C"]
P = {}
for i in range(3):
P[S[i]] = 1/3
ps = DiscreteProbabilitySpace(S,P)
ps.expectation() # 0.33
This is the random variable with
f("A") = 1,
f("B") = 2,
f("C") = 3,
for which the expectation is 2:
f = {}
for i in range(3):
f[S[i]] = i+1
rv = DiscreteRandomVariable(ps,f)
rv.expectation() # 2.00
On the other hand, I'd be happy with some syntax for creating a
random variable with the uniform distribution P(x) = 1/n on some
set or tuple. Currently this gives an error:
sage: rv = DiscreteRandomVariable(S,f)
but it could easily create the uniform distribution on S behind
the scenes. The reason I didn't do this is the following:
sage: ps1 = DiscreteProbabilitySpace(S,P)
sage: ps2 = DiscreteProbabilitySpace(S,P)
sage: ps1 == ps2
False
Probably this should be considered a bug. But to avoid creating
many copies of the "same" space, one should consider caching the
probability spaces. However, since dictionaries are mutable, it
can't be cached.
The correct solution might be to define additional types for the
uniform distribution and other standard distributions (which are
the most likely candidates to be created over and over) and have
only one instance of each.
My type DiscreteProbabilitySpace is also too naive -- it really
only implements a FiniteProbabilitySpace.
It is suitable for the examples I've taught in classical cryptography,
but for a statistics class one certainly needs infinite discrete
probability spaces and more classes for distributions on continuous
intervals.
Some thought also needs to be given to their ease of use. The
usual approach to this is to first write some documentation which
describes the desired behavior and then implement the classes and
functions.
--David
----- Forwarded message from "David R. Kohel" <[EMAIL PROTECTED]> -----
Date: Mon, 1 Dec 2008 09:04:12 +0100
From: "David R. Kohel" <[EMAIL PROTECTED]>
To: William Stein <[EMAIL PROTECTED]>
Cc: Paul Butler <[EMAIL PROTECTED]>
Subject: Re: Fwd: [sage-devel] Expected value of probability space
User-Agent: Mutt/1.5.6i
Dear William, Paul,
Indeed, the function definition should be:
def expectation(self):
r"""
The expectation of the discrete random variable, namely
$\sum_{x \in S} p(x) X[x]$,
where $X$ = self and $S$ is the probability space of $X$.
"""
E = 0
Omega = self.probability_space()
for x in self._function.keys():
E += Omega(x) * x
return E
rather than:
def expectation(self):
r"""
The expectation of the discrete random variable, namely
$\sum_{x \in S} p(x) X[x]$,
where $X$ = self and $S$ is the probability space of $X$.
"""
E = 0
Omega = self.probability_space()
for x in self._function.keys():
E += Omega(x) * self(x)
return E
Cheers,
David
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