# Use of random function in DensityDist

The code below (apologies for complexity) incorporates a random distribution on matrices defined using DensityDist. The matrices represent ways of transforming a set of n ordered objects into n singletons via a sequence of n-1 refinements. Hence the entries to matrices are positive integers and the sums of the rows are always n as the rows represent ordered partitions. The random method for the DensityDist construction is given by the function random_mx(). In the code below, this function is first tested using the stateless DensityDist.dist distribution, which gives correct results. However, when I run it from within a model context using the run_true_MCMC() function, the matrices generated are clearly incorrect (they have non-integer and negative entries, leaving aside the test values). Is this because the function random_mx() is not written in then?

``````import numpy as np
from pymc3 import *
from pymc3.distributions.multivariate import Dirichlet, Multinomial
from pymc3.distributions.dist_math import bound, logpow, factln
from pymc3.distributions.special import gammaln
import theano.tensor as tt
from theano import config
import warnings
import re
from scipy.special import binom

def random_mx():
"""Function to generate random matrix values."""
def replace_zero(s, j):
"""Replace jth 0 in string s with a 1. First character is given by j=0 etc."""
count = 0
for i in range(len(s)):
if s[i] == '0':
count += 1
if count == j + 1:
return (s[:i] + '1' + s[i + 1:])

def make_row(s):
"""Convert string of zeros (+) and ones (,) to matrix row, i.e. counting
partitions by size."""
c = re.split('1', s)
return [len(x) + 1 for x in c]

s = '0' * (n - 1)
result = list()
for i in range(n - 1):
num0 = n - 1 - i
j = np.random.choice(num0)
s = replace_zero(s, j)
row = make_row(s)
row = np.pad(row, (0, n - len(row)), mode='constant', constant_values=0)
result.append(row)
mx = np.array(result, dtype=np.int32)
return mx

def mx_logpdf():
return 0.0

def get_testval(n):
"""Get matrix testval."""
m1 = np.ones((n-1,n-1), dtype=np.int32)
m2 = np.tril(m1)
m3 = np.arange(1, n)
m4 = m3.reshape(-1,1)
m5 = np.flipud(m4)
return np.concatenate((m5, m2), axis=1)

def run_true_MCMC(sfs, prior, mut_rate, draws):
"""Define and run MCMC model for coalescent tree branch lengths."""
n = len(sfs) + 1
j_n = np.diag(1 / np.arange(2, n + 1))
sfs = np.array(sfs)
seg_sites = sum(sfs)
testval = get_testval(n)

with Model() as model:

def mx_logpdf(value=testval):
return 0.0

tmx = DensityDist('tmx', mx_logpdf, random=random_mx, shape=(n-1,n), testval=testval)
probs = Dirichlet('probs', prior, testval=prior)

tmx_print = tt.printing.Print('tmx')(tmx)
result = list()
for rowix in range(n - 1):
orp = tmx_print[rowix]
orp2 = tt.cast(orp, 'int32')
row = tt.bincount(orp2, minlength=n+1)
row = row[1:-1]
result.append(row)
mx = tt.stack(result, axis=0)
mx = tt.reshape(mx, (n-1,n-1))
mx = tt.transpose(mx)
jmx = tt.dot(mx, j_n)
conditional_probs = tt.dot(jmx, probs.T)

sfs_obs = Multinomial('sfs_obs', n=seg_sites, p=conditional_probs, observed=sfs)

with model:
step = Metropolis([probs, tmx])
trace = sample(5, step=step, tune=0, chains=1, progressbar=False)
print(summary(trace, varnames=['probs']))
return trace

sfs = [36, 4, 32, 0, 13, 0, 0]
n = len(sfs) + 1

#Generate a random matrix using DensityDist.dist. This gives a matrix of the desired form.
tmx = DensityDist.dist(mx_logpdf, random=random_mx)
print('Matrix generated by DensityDist.dist\n', tmx.random(), '\n')

#Now run the MCMC algorithm using DensityDist. The matrices now generated (excluding testval)
#do not have the correct form.
draws = 50000
mutation_rate = 5e-9
length = 1000
seq_mutation_rate = mutation_rate * length
prior = np.ones(n - 1) / (n - 1)
trace = run_true_MCMC(sfs, prior, seq_mutation_rate, draws)
``````

The problem in your case is that you have not implemented the logp correctly. Currently, it is implemented as

``````def mx_logpdf():
return 0.0
``````

But you need to rewrite it into a proper logp that evaluate on some input and gives a log(probability). I understand that you might not always be able to find a way to express your logp, in those cases you will not be able to use a MCMC method.

Hi Jungpen

Thanks for responding. The distribution is meant to be uniform (discrete), i.e. have the same logp for all variates. So I thought that any constant value should do, assuming that a constant multiple of probability is OK. If not, how should I achieve the uniform discrete distribution? Thanks, Helmut

It is discrete uniform but with constraint (ordered and sum to N for each row), I think the best way to implement this is to implement a chained transformation (in the continuous case we have ordered and sumto1 transformation)

I have had a further look at the documentation on transforms. I agree with you that the transform process seems to be the cause of my problem, particularly as the corresponding DensityDist.dist method works correctly. However, due to the nature of the algorithm that generates the desired support set of matrices, I do not think that chaining transforms is a feasible solution. Is it possible to force DensityDist to not refer to any transform, but simply use the function passed as the parameter â€˜randomâ€™?

Event if your random generation method is implemented correctly, it does not enforce constraint on your likelihood (they are often two completely different things!) So you still need to find a way to put the constrain one way (transformation) or the other (potential)

I have also looked at potential. I do not think either transforms or potentials will succeed as the valid matrices in my set are defined by a combinatorial algorithm, not numerical constraints. I am thinking of trying to implement the model in PyMC2 as this does not seem to have transforms etc. Probably using the decorator method to generate the appropriate stochastic random variable. What are your thoughts on this?

If you really cannot find a way to express the RV, using something like a SMC or ABC-SMC is more appropriate where you only need to supply a data generation algorithm.