Random sampling of sorted numbers within range

[vian]

Hi there,

I need your brains again for what I thought was a simple problem but turns out to be more complicated than I thought…

What I need is to be able to draw N numbers between 0 and 1, and they need to be sorted (any way). The following apparently works:

wtot = 1
b = pm.Dirichlet(‘b’, a=np.ones(n_sectors))
w = [pm.Deterministic(‘w_s{}’.format(i), wtot*b[i]) for i in range(n_sectors)]
for s in np.arange(1, n_sectors):
pm.Potential(‘prior_w_s{}’.format(s), pm.Uniform.dist(lower=0).logp(w[s-1]-w[s]))

(wtot is in case we wanna draw up to any number below 1, and it can be a random variable too, here I just set it to 1)

But I have 2 questions:

• I would assume the use of potential slows down how fast draws are accepted/rejected, is that right? In which case:
• Is there a better solution?

Thanks for your help!
Cheers,
MV

[junpenglao]

You might find some previous discussion helpful here:

Sampling uniformly in a triangular support

There are a couple of ways to do it: You can use the ordered transformation here. You can add jacobian correction for the volume change in your case above as well to make it proper. You can jointly sample from a bivariate standard Gaussian then rotating them by 45 degrees. The rotated sample is transformed into rates that lie in the unit square. See here. You can add a potential to restrict the order, see here cell [4] for an example

TL;dr:

tr = pm.transforms
Order = tr.Ordered()
Logodd = tr.LogOdds()
chain_tran = tr.Chain([Logodd, Order])

n = 5
with pm.Model() as m0:
    x = pm.Uniform("x", 0.0, 1.0, shape=n, transform=chain_tran,
                   testval=np.linspace(.1, .9, n))  # test value need to be ordered
    trace = pm.sample()

[vian]

Thanks! Right now I can’t make the Ordered transform work with Dirichlet (sum!=1) but that’s a really useful example for Uniform.

Cheers,
MV