# Error using Gibbs sampling for inference in a probabilistic graphical model

hi Dustin,

This is a follow up question to the one I asked yesterday on Gitter regarding implementing a graphical model with latent node in edward. I tried the Gibbs sampling for my problem as you suggested. Here is the code used

``````from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import tensorflow as tf
import numpy as np
import edward as ed
import matplotlib.pyplot as plt
import six
from time import time

from edward.models import InverseGamma, Normal, Empirical
from edward.util import get_blanket

"""
Directed Graphical model:
X->Y , Z->Y
X,Y,Z are assumed to be Gaussian distributions
data is observed on X,Y but not Z (latent node)

P(X,Y,Z) = P(Y|X,Z) * P(X)* P(Z)
P(Z) = N(loc=mu_z, scale=ig_z)
P(X) = N(loc=mu_x, scale=ig_x)
P(Y|X,Z) = N(loc=(b + w*X + wz*Z), scale=ig)

Estimate: mu_z,ig_z,mu_x,ig_x,b,w,wz
Data:  Y_train ,X_train

"""

def build_toy_dataset(N, b, w_o, w_h, noise_std=1):
D = len(w_o)+len(w_h)
w = np.concatenate((w_o,w_h))
x = np.zeros((N,D))
x[:,0] = np.random.normal(2, 4, size=N) # observed node
x[:,1] = np.random.normal(5, 6, size=N) # hidden node
y = np.dot(x, w) + np.ones(N)*b +np.random.normal(0, noise_std, size=N)
x_o = np.zeros((N,len(w_o)))
x_o[:,0] = x[:,0]
return x_o, y

#ed.set_seed(42)

N = 1000  # number of data points
D = 1  # number of observed nodes
Dz = 1 # number of hidden nodes

# DATA
w_true = np.ones(D) * 5.0
w_hid = np.ones(D) * 3.0
X_train, y_train = build_toy_dataset(N, 1, w_true, w_hid)
#X_test, y_test = build_toy_dataset(N,1, w_true)

# MODEL

#X = tf.placeholder(tf.float32, [N, D])

## Latent node - Z
alpha_z = tf.Variable(0.5, trainable=False)
beta_z = tf.Variable(0.7, trainable=False)
ig_z = InverseGamma(alpha_z, beta_z)
mu_z = Normal(loc=0.0, scale=1.0)
z = Normal(loc=tf.ones(N) * mu_z, scale=tf.ones([N]) * tf.sqrt(ig_z))

## Observed node - X
alpha_x = tf.Variable(0.5, trainable=False)
beta_x = tf.Variable(0.7, trainable=False)
ig_x = InverseGamma(alpha_x, beta_x)
mu_x = Normal(loc=0.0, scale=1.0)
X = Normal(loc=tf.ones(N) * mu_x, scale=tf.ones([N]) * tf.sqrt(ig_x))

# Observed node - Y
alpha = tf.Variable(0.5, trainable=False)
beta = tf.Variable(0.7, trainable=False)
ig = InverseGamma(alpha, beta)
w = Normal(loc=tf.zeros(D), scale=tf.ones(D))
wz = Normal(loc=tf.zeros(Dz), scale=tf.ones(Dz))
b = Normal(loc=tf.zeros(1), scale=tf.ones(1))
y = Normal(loc=(X*w+z*wz+b), scale=tf.ones([N]) * tf.sqrt(ig))

# INFERENCE

T = 500 # number of MCMC samples

# Conditionals
muz_cond = ed.complete_conditional(mu_z)
igz_cond = ed.complete_conditional(ig_z)
mux_cond = ed.complete_conditional(mu_x)
igx_cond = ed.complete_conditional(ig_x)
ig_cond = ed.complete_conditional(ig)

b_cond = ed.complete_conditional(b)
w_cond = ed.complete_conditional(w)
wz_cond = ed.complete_conditional(wz)

sess = ed.get_session()

# Initialize randomly
muz_est,igz_est,w_est,wz_est,ig_est,b_est = sess.run([mu_z, ig_z, w, wz, ig, b])

print('Initial parameters:')
print(muz_est,igz_est,w_est,wz_est,ig_est,b_est)

# Gibbs sampler
cond_dict = {mu_z:muz_est, ig_z:igz_est, w:w_est, wz:wz_est, ig:ig_est, b:b_est, X: X_train, y: y_train}
t0 = time()
T = 500
for t in range(T):
muz_est = sess.run(muz_cond, cond_dict)
cond_dict[mu_z] = muz_est
igz_est = sess.run(igz_cond, cond_dict)
cond_dict[ig_z] = igz_est

w_est,wz_est,ig_est,b_est = sess.run([w_cond,wz_cond,ig_cond,b_cond], cond_dict)
cond_dict[w] = w_est
cond_dict[wz] = wz_est
cond_dict[ig] = ig_est
cond_dict[b] = b_est

print('took %.3f seconds to run %d iterations' % (time() - t0, T))

print()
print('Final sample for parameters::')
print(muz_est,igz_est,w_est,wz_est,ig_est,b_est)
``````

but I get the following error:

``````'sufficient statistics.' % str(dist_key))
NotImplementedError: Conditional distribution has sufficient statistics (('#Add', ('#Mul', (2.0,),
(<tf.Tensor 'Normal_10/sample/Reshape:0' shape=(1000,) dtype=float32>,),
(<tf.Tensor 'Normal_13/sample/Reshape:0' shape=(1,) dtype=float32>,),
('#x',)), ('#Mul', (2.0,), (<tf.Tensor 'Normal_4/sample/Reshape:0' shape=(1000,) dtype=float32>,),
(<tf.Tensor 'Normal_16/sample/Reshape:0' shape=(1,) dtype=float32>,), ('#x',)), ('#CPow2.0000e+00',
('#x',))), ('#CPow2.0000e+00', ('#x',)), ('#x',)), but no available exponential-family distribution has those sufficient statistics.
``````

The error indicates that the full conditionals are not able to be computed for b_cond, w_cond, wz_cond. I think this is because of the definition of loc in

``````y = Normal(loc=(X*w+z*wz+b), scale=tf.ones([N]) * tf.sqrt(ig))
``````

However, this is similar to a Bayesian linear regression, so I defined the conjugate priors for b,w,wz accordingly. Do you have any insights on why this is happening and if I should define the complete conditionals for these variable separately?

I have the same issue, the same model works fine in pymc3, but it raise the error

`````` Conditional distribution has sufficient statistics
``````

in Edward

As you note, the error says that the symbolic algebra behind `complete_conditional` isn’t smart enough to handle those cases.

If you’d like to use Gibbs sampling, you can still work out the algebra yourself and define your own complete conditional distributions. Then pass them into Gibbs sampling. For example,

``````ed.Gibbs({mu: qmu, b: qb}, proposal_vars={mu: mu_cond, b: b_cond}, data={x: x_data, y: y_data})
``````

where `mu_cond` and `b_cond` are complete conditional distributions p(mu | b, x, y) and p(b | mu, x, y).

Alternatively, contributions are welcome to make `ed.complete_conditional` smarter (check the `conjugacy` source code).

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