def mv_simple():
mu = np.array([-.1, .5, 1.1])
p = np.array([
[2., 0, 0],
[.05, .1, 0],
[1., -0.05, 5.5]])
tau = np.dot(p, p.T)
with pm.Model() as model:
pm.MvNormal('x', pm.constant(mu), pm.constant(tau), shape=3, testval=np.array([.1, 1., .8]))
H = tau
C = np.linalg.inv(H)
return model.test_point, model, (mu, C)
def mv_simple():
mu = np.array([-.1, .5, 1.1])
p = np.array([
[2., 0, 0],
[.05, .1, 0],
[1., -0.05, 5.5]])
tau = np.dot(p, p.T)
with pm.Model() as model:
x = pm.MvNormal('x', pm.constant(mu), pm.constant(
tau), shape=3, testval=np.array([.1, 1., .8]))
H = tau
C = np.linalg.inv(H)
return model.test_point, model, (mu, C)
def mv_simple_discrete():
d = 2
n = 5
p = np.array([.15, .85])
with pm.Model() as model:
pm.Multinomial('x', n, pm.constant(p), shape=d, testval=np.array([1, 4]))
mu = n * p
# covariance matrix
C = np.zeros((d, d))
for (i, j) in product(range(d), range(d)):
if i == j:
C[i, i] = n * p[i] * (1 - p[i])
else:
C[i, j] = -n * p[i] * p[j]
return model.test_point, model, (mu, C)
def mv_simple_discrete():
d= 2
n = 5
p = np.array([.15,.85])
with pm.Model() as model:
x = pm.Multinomial('x', n, pm.constant(p), shape=d, testval=np.array([1,4]))
mu = n * p
#covariance matrix
C = np.zeros((d,d))
for (i, j) in product(range(d), range(d)):
if i == j:
C[i,i] = n * p[i]*(1-p[i])
else:
C[i,j] = -n*p[i]*p[j]
return model.test_point, model, (mu, C)
ndata = 100
spread = 3
centers = np.array([-spread, 0, spread])
p = stick_breaking(alpha=10.0, k=30)
# simulate data from mixture distribution
v = np.random.randint(0, k, ndata)
data = centers[v] + np.random.randn(ndata)
# plt.hist(data)
# plt.show()
model = pm.Model()
with model:
# cluster sizes
a = pm.constant(np.array([1., 1., 1.]))
# p = pm.Dirichlet('p', a=a, shape=k)
# ensure all clusters have some points
# p_min_potential = pm.Potential('p_min_potential', tt.switch(tt.min(p) < .1, -np.inf, 0))
# cluster centers
means = pm.Normal('means', mu=[0, 0, 0], sd=15, shape=k)
# break symmetry
# order_means_potential = pm.Potential('order_means_potential',
# tt.switch(means[1]-means[0] < 0, -np.inf, 0)
# + tt.switch(means[2]-means[1] < 0, -np.inf, 0))
# measurement error
sd = pm.Uniform('sd', lower=0, upper=20)
# latent cluster of each observation
# priors
mu = pm.Normal('mu', mu=0, tau=1/100**2, shape=(2,1))
lmbda = pm.Gamma('lambda', alpha=0.001, beta=0.001, shape=(2,1))
r = pm.Uniform('r', lower=-1, upper=1)
sigma = pm.Deterministic('sigma', tt.sqrt(1/lmbda))
# Reparameterization
#FIXME: How to create (and then inverse) a simple 2x2 matrix???
T = tt.stacklists([[1/lmbda[0] , r*sigma[0]*sigma[1]],
[r*sigma[0]*sigma[1], 1/lmbda[1]]])
# T = tt.stack([1/lmbda[0] , r*sigma[0]*sigma[1],
# r*sigma[0]*sigma[1], 1/lmbda[1]])
# TI = tt.invert(T)
# TI = tt.matrix(T)
# TODO? Side-step inversion by doing it myself, i.e. 1/det(A)*reshuffle(A)?
testtau = pm.constant(np.eye(2)) # works...
pm.det(testtau) # works...
x = pm.MvNormal('x', mu=0, tau=testtau)
# # Reparameterization
# sigma[1] <- 1/sqrt(lambda[1])
# sigma[2] <- 1/sqrt(lambda[2])
# T[1,1] <- 1/lambda[1]
# T[1,2] <- r*sigma[1]*sigma[2]
# T[2,1] <- r*sigma[1]*sigma[2]
# T[2,2] <- 1/lambda[2]
# TI[1:2,1:2] <- inverse(T[1:2,1:2])
# data come from a Gaussian
# x = pm.Normal('x', mu=mu, sd=sigma, observed=x)
#model{
# # Observed Returns
# for (i in 1:m){
# k[i] ~ dbin(theta,n)
# }
# # Priors on Rate Theta and Number n
# theta ~ dbeta(1,1)
# n ~ dcat(p[])
# for (i in 1:nmax){
# p[i] <- 1/nmax
# }
#}
with model:
# Priors on rate theta and number n
theta = pm.Beta('theta', alpha=1, beta=1)
p = pm.constant(np.ones(nmax)/nmax)
n = pm.Categorical('n', p=p, shape=1) #FIXME: How to use this properly?
# Observed Returns
# k = pm.Binomial('k', p=theta, n=n, observed=k, shape=m)
# instantiate samplers
values_np = np.ones(nmax)/nmax #FIXME: How to use this properly?
step1 = pm.Metropolis([theta])
step2 = pm.ElemwiseCategoricalStep(var=n, values=values_np)
stepFunc = [step1, step2]
# draw posterior samples (in 4 parallel running chains), TODO: very slow!?
Nsample = 100
Nchains = 4
traces = pm.sample(Nsample, step=stepFunc, njobs=Nchains)
plotVars = ['theta','n']
axs = pm.traceplot(traces, vars=plotVars, combined=False)
spread = 3
centers = np.array([-spread, 0, spread])
p=stick_breaking(alpha=10.0,k=30)
# simulate data from mixture distribution
v = np.random.randint(0, k, ndata)
data = centers[v] + np.random.randn(ndata)
# plt.hist(data)
# plt.show()
model = pm.Model()
with model:
# cluster sizes
a = pm.constant(np.array([1., 1., 1.]))
# p = pm.Dirichlet('p', a=a, shape=k)
# ensure all clusters have some points
# p_min_potential = pm.Potential('p_min_potential', tt.switch(tt.min(p) < .1, -np.inf, 0))
# cluster centers
means = pm.Normal('means', mu=[0, 0, 0], sd=15, shape=k)
# break symmetry
# order_means_potential = pm.Potential('order_means_potential',
# tt.switch(means[1]-means[0] < 0, -np.inf, 0)
# + tt.switch(means[2]-means[1] < 0, -np.inf, 0))
# measurement error
sd = pm.Uniform('sd', lower=0, upper=20)
#model{
# # Observed Returns
# for (i in 1:m){
# k[i] ~ dbin(theta,n)
# }
# # Priors on Rate Theta and Number n
# theta ~ dbeta(1,1)
# n ~ dcat(p[])
# for (i in 1:nmax){
# p[i] <- 1/nmax
# }
#}
with model:
# Priors on rate theta and number n
theta = pm.Beta('theta', alpha=1, beta=1)
p = pm.constant(np.ones(nmax) / nmax)
n = pm.Categorical('n', p=p, shape=1) #FIXME: How to use this properly?
# Observed Returns
# k = pm.Binomial('k', p=theta, n=n, observed=k, shape=m)
# instantiate samplers
values_np = np.ones(nmax) / nmax #FIXME: How to use this properly?
step1 = pm.Metropolis([theta])
step2 = pm.ElemwiseCategoricalStep(var=n, values=values_np)
stepFunc = [step1, step2]
# draw posterior samples (in 4 parallel running chains), TODO: very slow!?
Nsample = 100
Nchains = 4
traces = pm.sample(Nsample, step=stepFunc, njobs=Nchains)
plotVars = ['theta', 'n']
axs = pm.traceplot(traces, vars=plotVars, combined=False)
