[Numpy-discussion] Ticket #1223...

[David Goldsmith]

On Tue, Jun 29, 2010 at 8:16 PM, Bruce Southey <bsouthey at gmail.com> wrote:

> On Tue, Jun 29, 2010 at 6:03 PM, David Goldsmith
> <d.l.goldsmith at gmail.com> wrote:
> > On Tue, Jun 29, 2010 at 3:56 PM, <josef.pktd at gmail.com> wrote:
> >>
> >> On Tue, Jun 29, 2010 at 6:37 PM, David Goldsmith
> >> <d.l.goldsmith at gmail.com> wrote:
> >> > ...concerns the behavior of numpy.random.multivariate_normal; if
> that's
> >> > of
> >> > interest to you, I urge you to take a look at the comments (esp. mine
> >> > :-) );
> >> > otherwise, please ignore the noise.  Thanks!
> >>
> >> You should add the link to the ticket, so it's faster for everyone to
> >> check what you are talking about.
> >>
> >> Josef
> >
> > Ooops!  Yes I should; here it is:
> >
> > http://projects.scipy.org/numpy/ticket/1223
> > Sorry, and thanks, Josef.
> >
> > DG
> >
> >
> > _______________________________________________
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> > NumPy-Discussion at scipy.org
> > http://mail.scipy.org/mailman/listinfo/numpy-discussion
> >
> >
> As I recall, there is no requirement for the variance/covariance of
> the normal distribution to be positive definite.
>

No, not positive definite, positive *semi*-definite: yes, the variance may
be zero (the cov may have zero-valued eigenvalues), but the claim (and I
actually am "neutral" about it, in that I wanted to reference the claim in
the docstring and was told that doing so was unnecessary, the implication
being that this is a "well-known" fact), is that, in essence (in 1-D) the
variance can't be negative, which seems clear enough.  I don't see you
disputing that, and so I'm uncertain as to how you feel about the proposal
to "weakly" enforce symmetry and positive *semi*-definiteness.  (Now, if you
dispute that even requiring positive *semi*-definiteness is desirable,
you'll have to debate that w/ some of the others, because I'm taking their
word for it that indefiniteness is "unphysical.")

DG

>From http://en.wikipedia.org/wiki/Multivariate_normal_distribution
"The covariance matrix is allowed to be singular (in which case the
corresponding distribution has no density)."

So you must be able to draw random numbers from such a distribution.
Obviously what those numbers really mean is another matter (I presume
the dependent variables should be a linear function of the independent
variables) but the user *must* know since they entered it. Since the
function works the docstring Notes comment must be wrong.

Imposing any restriction means that this is no longer a multivariate
normal random number generator. If anything, you can only raise a
warning about possible non-positive definiteness but even that will
vary depending how it is measured and on the precision being used.


Bruce
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-- 
Mathematician: noun, someone who disavows certainty when their uncertainty
set is non-empty, even if that set has measure zero.

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lies, prevents mankind from committing a general suicide.  (As interpreted
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