def st_d_logp(x, mu, nu, sigma2):
x_p = (x - mu) / T.sqrt(sigma2)
prob = T.log(
T.gamma((nu + 1.0) / 2.0) /
(T.gamma(nu / 2.0) * T.sqrt(pi * nu * sigma2)) *
T.power(1.0 + x_p**2 / nu, -(nu + 1) / 2.0))
return prob
def likelihood(xs):
return T.sum(
T.log(beta) -
T.log(2.0 * std *
T.sqrt(T.gamma(1. / beta) / T.gamma(3. / beta))) -
T.gammaln(1.0 / beta) + -T.power(
T.abs_(xs - mu) / std *
T.sqrt(T.gamma(1. / beta) / T.gamma(3. / beta)), beta))
def likelihood(xs):
return tt.sum(
tt.log(beta) -
tt.log(2.0 * std *
tt.sqrt(tt.gamma(1. / beta) / tt.gamma(3. / beta))) -
tt.gammaln(1.0 / beta) + -tt.power(
tt.abs_(xs - mu) / std *
tt.sqrt(tt.gamma(1. / beta) / tt.gamma(3. / beta)), beta))
def kl_div_ng_ng(p_alpha, p_beta, p_nu, p_mu, q_alpha, q_beta, q_nu, q_mu):
kl_dist = 1.0 / 2.0 * p_alpha / p_beta * (q_mu - p_mu)**2.0 * q_nu
kl_dist = kl_dist + 1.0 / 2.0 * q_nu / p_nu
kl_dist = kl_dist - 1.0 / 2.0 * T.log(q_nu / p_nu)
kl_dist = kl_dist - 1.0 / 2.0 + q_alpha * T.log(p_beta / q_beta) - T.log(
T.gamma(p_alpha) / T.gamma(q_alpha))
kl_dist = kl_dist + (p_alpha - q_alpha) * special_functions.psi(
p_alpha) - (p_beta - q_beta) * p_alpha / p_beta
return kl_dist
def __init__(self, input, n_in, n_out, W=None, b=None):
LogisticRegression.__init__(self, input, n_in, n_out, W=None, b=None)
lin_out = T.dot(input, self.W) + self.b
n = n_out - 1
k = np.arange(n_out)
binom = (T.log(T.gamma(n + 1)) - T.log(T.gamma(k + 1)) -
T.log(T.gamma(n - k + 1)))
pre_softmax = -(n - k) * lin_out - n * T.nnet.softplus(-lin_out)
self.p_y_given_x = T.nnet.softmax(binom + pre_softmax)
self.y_pred = T.argmax(self.p_y_given_x, axis=1)
def kl_div_dir_dir(p_a0, q_a0):
a0 = T.sum(p_a0, axis=1)
b0 = T.sum(q_a0, axis=1)
#assume a0 = b0+1, since there was only one observation added
kl_dist = T.log(T.gamma(a0) / T.gamma(b0)) + T.sum(
T.log(T.gamma(q_a0) / T.gamma(p_a0)), axis=1)
# kl_dist = T.log(b0)-T.log(T.sum((p_a0-q_a0)*q_a0))
kl_dist = kl_dist + T.sum(
(p_a0 - q_a0) *
(special_functions.psi(p_a0) - special_functions.psi(a0).reshape(
(a0.shape[0], 1))),
axis=1)
# kl_dist = kl_dist + T.sum((p_a0-q_a0)*(special_functions.psi(p_a0)-special_functions.psi(a0)),axis=1)
return kl_dist
def binomial_elemwise(self, y, t):
# specify broadcasting dimensions (multiple inputs to multiple
# density estimations)
if self.convolutional:
est_shuf = y.dimshuffle(0, 1, 2, 3)
v_shuf = t.dimshuffle(0, 1, 2, 3)
else:
est_shuf = y.dimshuffle(0, 1)
v_shuf = t.dimshuffle(0, 1)
# Calculate probabilities of current v's to be sampled given estimations
# Real-valued observation -> Binomial distribution
# Binomial coefficient (factorial(x) == gamma(x+1))
bin_coeff = 1 / (T.gamma(1 + v_shuf) * T.gamma(2 - v_shuf))
pw_probs = bin_coeff * T.pow(est_shuf, v_shuf) * T.pow(
1. - est_shuf, 1. - v_shuf)
return pw_probs
def logp(self, value):
"""
Calculate log-probability of EFF distribution at specified value.
Parameters
----------
value : numeric
Value(s) for which log-probability is calculated. If the log probabilities for multiple
values are desired the values must be provided in a numpy array or theano tensor
Returns
-------
TensorVariable
"""
gamma = self.gamma
x = (self.location - value) / self.scale
cte = tt.sqrt(
np.pi) * self.scale * tt.gamma(gamma - 0.5) / tt.gamma(gamma)
log_d = -gamma * tt.log(1. + x**2) - tt.log(cte)
return log_d
def _Ps(self, n, l):
return T.exp(-l) * (l ** n) / T.gamma(n + 1)
def theano_beta_fn(a, b):
return T.gamma(a) * T.gamma(b) / T.gamma(a+b)
data = np.asarray((exampleData[0:q], exampleDataNLost[0:q]))
with pm.Model() as BdWwithcfromNorm:
alpha = pm.Uniform('alpha', 0.0001, 1000.0, testval=1.0)
beta = pm.Uniform('beta', 0.0001, 1000.0, testval=1.0)
# If c=1, the dW collapses to a geometric distribution. We assume
# that in a usual case, customer survival probability normally
# stay the same over time.
# If mu = 1.5 we get exactly the same sampling results as if
# we had used a uniform prior. The result is slightly better than
# defining a N(1,None) variable.
c = pm.Normal('c', mu=0.5, testval=1.0)
p = [0.] * n
s = [1.] * n
logB = tt.gamma(alpha + beta) / tt.gamma(beta)
for t in range(1, n):
pt1 = tt.gamma(beta + (t - 1)**c) / tt.gamma(beta + alpha + (t - 1)**c)
pt2 = tt.gamma(beta + t**c) / tt.gamma(beta + alpha + t**c)
s[t] = pt2 * logB
p[t] = s[t - 1] - s[t]
p = tt.stack(p, axis=0)
s = tt.stack(s, axis=0)
# Log-likelihood function
def logp(data):
observedRenewed = data[0, :]
observedReleased = data[1, :]
# Released entries every year
released = tt.mul(p[1:].log(), observedReleased[1:])
def __init__(self,
rng,
input,
batch_size,
in_size,
latent_size,
W_a=None,
W_b=None,
epsilon=0.01):
self.srng = theano.tensor.shared_randomstreams.RandomStreams(
rng.randint(999999))
self.input = input
# setup variational params
if W_a is None:
W_values = np.asarray(
0.01 * rng.standard_normal(size=(in_size, latent_size - 1)),
dtype=theano.config.floatX)
W_a = theano.shared(value=W_values, name='W_a')
if W_b is None:
W_values = np.asarray(
0.01 * rng.standard_normal(size=(in_size, latent_size - 1)),
dtype=theano.config.floatX)
W_b = theano.shared(value=W_values, name='W_b')
self.W_a = W_a
self.W_b = W_b
# compute Gamma samples
uniform_samples = T.cast(
self.srng.uniform(size=(batch_size, latent_size - 1),
low=0.01,
high=0.99), theano.config.floatX)
self.a = Softplus(T.dot(self.input, self.W_a))
self.b = Softplus(T.dot(self.input, self.W_b))
v_samples = ((uniform_samples * self.a * T.gamma(self.a))
**(1 / self.a)) / self.b
# setup variables for recursion
stick_segment = theano.shared(value=np.zeros(
(batch_size, ), dtype=theano.config.floatX),
name='stick_segment')
remaining_stick = theano.shared(value=np.ones(
(batch_size, ), dtype=theano.config.floatX),
name='remaining_stick')
def compute_latent_vars(i, stick_segment, remaining_stick, v_samples):
# compute stick segment
stick_segment = v_samples[:, i] * remaining_stick
remaining_stick *= (1 - v_samples[:, i])
return (stick_segment, remaining_stick)
(stick_segments, remaining_sticks), updates = theano.scan(
fn=compute_latent_vars,
outputs_info=[stick_segment, remaining_stick],
sequences=T.arange(latent_size - 1),
non_sequences=[v_samples],
strict=True)
self.avg_used_dims = T.mean(T.sum(remaining_sticks > epsilon, axis=0))
self.latent_vars = T.transpose(
T.concatenate([
stick_segments,
T.shape_padaxis(remaining_sticks[-1, :], axis=1).T
],
axis=0))
self.params = [self.W_a, self.W_b]