def gibbs(y,
N=None,K=None,alpha=1,
iters=500,burnin=0,skip=0,
truth=None):
"""
gibbs sampling for a stochastic block model
N = |nodes|
K = |classes|
N2 = |edges| = N^2 - N = 2 * (N-1 + ... 1)
in my notation,
x" means "curr x"
"x_" means "next x"
pi : [0,1]^K
pi[k] : [0,1]
z : {1..K}^N
z[i] : {1..K}
W : [0,1]^(K^2)
W[k,l] : [0,1]
y : {0,1}^|y|
0 <= |y| <= N2
"""
print '--- Gibbs Sampling ---'
assert N and K
assert 0 <= burnin < iters and 0 <= skip < iters
assert truth is not None
N2 = N**2 - N
probs, rands = zeros(iters), zeros(iters)
true_z = truth
# Init: sample from priors
# dirichlet prior
alpha = ones(K,dtype=int)
pi_ = sample.dirichlet(ones(K))
pi = pi_.copy()
# beta prior
h,t = ones((K,K)), ones((K,K))
W_ = sample.beta(1,1, size=(K,K))
W = W_.copy()
# categorical prior
z_ = nans(N,dtype=int)
for i in xrange(N):
z_[i] = sample.categorical(pi)
z = z_.copy()
pis, Ws, zs = [pi],[W],[z]
# Iter: sample from posteriors
for it in xrange(iters):
if (it+1)%100==0: print '- %d -' % (it+1)
for k in xrange(K): alpha[k] += sum(z==k)
pi_ = sample.dirichlet(alpha)
#print alpha
for (k,l) in cross(K,K):
Ykl = array([y[i,j] for (i,j) in cross(N,N)
if i!=j and y[i,j]!=nan and z[i]==k and z[j]==l])
h[k,l] += sum(Ykl==1)
t[k,l] += sum(Ykl==0)
W_ = sample.beta(h,t, size=(K,K))
#print h,t
for i in xrange(N):
zP = array([pi[zi]
* prod([W[zi,z[j]] if y[i,j] else 1-W[zi,z[j]]
for j in xrange(N)
if j!=i and y[i,j]!=nan])
* prod([W[z[j],zi] if y[j,i] else 1-W[z[j],zi]
for j in xrange(N)
if j!=i and y[j,i]!=nan])
for zi in xrange(K)])
zP /= sum(zP)
z_[i] = sample.categorical(zP)
#print z
pi = pi_.copy()
W = W_.copy()
z = z_.copy()
pis.append(pi)
Ws.append(W)
zs.append(z)
# compute log-probability; should (non-monotonically) increase
Ber = sum(log(W[z[i],z[j]] if y[i,j] else 1-W[z[i],z[j]])
for (i,j) in cross(N,N) if i!=j and y[i,j]!=nan)
Cat = sum(log(pi[z[i]]) for i in range(N))
Dir = sum((alpha[k]-1) * log(pi[k])
for k in xrange(K)) - log_Beta(alpha)
probs[it] = Ber + Cat + Dir
# compute rand_index by samples
rands[it] = rand_index(true_z, z)
return pis,zs,Ws, probs,rands
def gibbs(y, N=None,K=None,alpha=1, iters=500,burnin=0,skip=0):
"""
gibbs sampling for a stochastic block model
N = |nodes|
K = |classes|
N2 = |edges| = N^2 - N = 2 * (N-1 + ... 1)
in my notation,
x" means "curr x"
"x_" means "next x"
pi : [0,1]^K^N
pi[i] : [0,1]^K
s : {1..K}^|edges|
s[ij] : {1..K}
r : {1..K}^|edges|
r[ij] : {1..K}
W : [0,1]^(K^2)
W[k,l] : [0,1]
y : {0,1}^|y|
0 <= |y| <= |edges|
"""
print '--- Gibbs Sampling ---'
assert N and K
N2 = N**2 - N
probs = nans(iters)
# Init: sample from priors
# dirichlet prior
alpha = ones((N,K),dtype=int)
pi_ = sample.dirichlet(ones(K), size=N)
pi = pi_.copy()
# beta prior
h,t = ones((K,K)), ones((K,K))
W_ = sample.beta(1,1, size=(K,K))
W = W_.copy()
# categorical prior
r_ = nans((N,N))
for (i,j) in cross(N,N):
if i==j: continue
r_[i,j] = sample.categorical(pi[i])
r = r_.copy()
# categorical prior
s_ = nans((N,N))
for (i,j) in cross(N,N):
if i==j: continue
s_[i,j] = sample.categorical(pi[j])
s = s_.copy()
pis,Ws,rs,ss = [pi],[W],[r],[s]
# Iter: sample from posteriors
for it in range(iters):
if (it+1)%100==0: print '- %d -' % (it+1)
for i in range(N):
for k in range(K):
alpha[i,k] += sum(r[i,:]==k) + sum(s[:,i]==k)
pi_[i] = sample.dirichlet(alpha[i])
for (k,l) in cross(K,K):
h[k,l] += sum((y==1) * (r==k) * (s==l))
t[k,l] += sum((y==0) * (r==k) * (s==l))
W_ = sample.beta(h,t, size=(K,K))
r_ = nans((N,N))
for (i,j) in cross(N,N):
if i==j: continue
l = s[i,j]
rP = array([pi[i][k] * (W[k,l] if y[i,j] else 1-W[k,l])
for k in range(K)])
rP /= sum(rP)
r_[i,j] = sample.categorical(rP)
s_ = nans((N,N))
for (i,j) in cross(N,N):
if i==j: continue
k = r[i,j]
sP = array([pi[j][l] * (W[k,l] if y[i,j] else 1-W[k,l])
for l in range(K)])
sP /= sum(sP)
s_[i,j] = sample.categorical(sP)
pi = pi_.copy()
W = W_.copy()
r = r_.copy()
s = s_.copy()
pis.append(pi)
Ws.append(W)
rs.append(r)
ss.append(s)
# compute log-probability; should (non-monotonically) increase
Ber = sum(log(W[r[i,j], s[i,j]] if y[i,j] else 1-W[r[i,j], s[i,j]])
for (i,j) in cross(N,N) if i!=j and y[i,j]!=nan)
Cat = sum(log(pi[i][r[i,j]]) +
log(pi[j][s[i,j]])
for (i,j) in cross(N,N) if i!=j)
Dir = sum(sum((alpha[i][k]-1) * log(pi[i][k]) for k in range(K))
- log_Beta(alpha[i])
for i in range(N))
probs[it] = Ber + Cat + Dir
return pis,rs,ss,Ws, probs