def marginal_likelihood(self,
name,
X,
y,
noise,
jitter=0.0,
is_observed=True,
**kwargs):
R"""
Returns the marginal likelihood distribution, given the input
locations `X` and the data `y`.
This is integral over the product of the GP prior and a normal likelihood.
.. math::
y \mid X,\theta \sim \int p(y \mid f,\, X,\, \theta) \, p(f \mid X,\, \theta) \, df
Parameters
----------
name: string
Name of the random variable
X: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
y: array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise: scalar, Variable, or Covariance
Standard deviation of the Gaussian noise. Can also be a Covariance for
non-white noise.
jitter: scalar
A small correction added to the diagonal of positive semi-definite
covariance matrices to ensure numerical stability. Default value is 0.0.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
if not isinstance(noise, Covariance):
noise = pm.gp.cov.WhiteNoise(noise)
mu, cov = self._build_marginal_likelihood(X, noise, jitter)
self.X = X
self.y = y
self.noise = noise
if is_observed:
return pm.MvNormal(name, mu=mu, cov=cov, observed=y, **kwargs)
else:
warnings.warn(
"The 'is_observed' argument has been deprecated. If the GP is "
"unobserved use gp.Latent instead.",
FutureWarning,
)
return pm.MvNormal(name, mu=mu, cov=cov, **kwargs)
def conditional(self, name, Xnew, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
`Xnew` will be split by columns and fed to the relevant
covariance functions based on their `input_dim`. For example, if
`cov_func1`, `cov_func2`, and `cov_func3` have `input_dim` of 2,
1, and 4, respectively, then `Xnew` must have 7 columns and a
covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
The distribution returned by `conditional` does not have a
Kronecker structure regardless of whether the input points lie
on a full grid. Therefore, `Xnew` does not need to have grid
structure.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, size=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, given=None, **kwargs):
R"""
Returns the approximate conditional distribution of the GP evaluated over
new input locations `Xnew`.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise: bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given: dict
Can optionally take as key value pairs: `X`, `Xu`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, size=shape, **kwargs)
def test_simultaneous_size_and_dims(self, with_dims_ellipsis):
with pm.Model() as pmodel:
x = pm.ConstantData("x", [1, 2, 3], dims="ddata")
assert "ddata" in pmodel.dim_lengths
# Size does not include support dims, so this test must use a dist with support dims.
kwargs = dict(name="y", size=2, mu=at.ones((3, 4)), cov=at.eye(4))
if with_dims_ellipsis:
y = pm.MvNormal(**kwargs, dims=("dsize", ...))
assert pmodel.RV_dims["y"] == ("dsize", None, None)
else:
y = pm.MvNormal(**kwargs, dims=("dsize", "ddata", "dsupport"))
assert pmodel.RV_dims["y"] == ("dsize", "ddata", "dsupport")
assert "dsize" in pmodel.dim_lengths
assert y.eval().shape == (2, 3, 4)
def conditional(self, name, Xnew, given=None, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that
the GP prior was over, the conditional distribution over a
set of new points, `f_*` is
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) K(X, X)^{-1} f \,,
K(X_*, X_*) - K(X_*, X) K(X, X)^{-1} K(X, X_*) \right)
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values.
given: dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, *givens)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, size=shape, **kwargs)
def model():
# Priors
sigma_y = pymc.Uniform('sigma_y', lower=0, upper=100)
tau_y = pymc.Lambda('tau_y', lambda s=sigma_y: s**-2)
xi = pymc.Uniform('xi', lower=0, upper=100, value=np.zeros(K))
mu_raw = pymc.Normal('mu_raw', mu=0., tau=0.0001, value=np.zeros(K))
Tau_B_raw = pymc.Wishart('Tau_B_raw', df, Tau=np.diag(np.ones(K)))
B_raw = pymc.MvNormal('B_raw', mu_raw, Tau_B_raw, value=np.zeros((J, K)))
# Model
@pymc.deterministic(plot=True)
def B(xi=xi, B_raw=B_raw):
return xi * B_raw
@pymc.deterministic
def mu(xi=xi, mu_raw=mu_raw):
return xi * mu_raw
@pymc.deterministic(plot=False)
def y_hat(B=B, X=X, i=index_c):
return np.sum(B[i, ] * X, axis=1)
# Likelihood
@pymc.stochastic(observed=True)
def y_i(value=y, mu=y_hat, tau=tau_y):
return pymc.normal_like(value, mu, tau)
return vars()
def conditional(self, name, Xnew, pred_noise=False, given=None, jitter=0.0, **kwargs):
R"""
Returns the approximate conditional distribution of the GP evaluated over
new input locations `Xnew`.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise: bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given: dict
Can optionally take as key value pairs: `X`, `Xu`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC for more information.
jitter: scalar
A small correction added to the diagonal of positive semi-definite
covariance matrices to ensure numerical stability. For conditionals
the default value is 0.0.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens, jitter)
return pm.MvNormal(name, mu=mu, cov=cov, **kwargs)
def test_mv_missing_data_model(self):
data = ma.masked_values([[1, 2], [2, 2], [-1, 4], [2, -1], [-1, -1]],
value=-1)
model = pm.Model()
with model:
mu = pm.Normal("mu", 0, 1, size=2)
sd_dist = pm.HalfNormal.dist(1.0)
chol, *_ = pm.LKJCholeskyCov("chol_cov",
n=2,
eta=1,
sd_dist=sd_dist,
compute_corr=True)
y = pm.MvNormal("y", mu=mu, chol=chol, observed=data)
inference_data = pm.sample(100,
chains=2,
return_inferencedata=True)
# make sure that data is really missing
assert isinstance(y.owner.op,
(AdvancedIncSubtensor, AdvancedIncSubtensor1))
test_dict = {
"posterior": ["mu", "chol_cov"],
"observed_data": ["y"],
"log_likelihood": ["y"],
}
fails = check_multiple_attrs(test_dict, inference_data)
assert not fails
def test_get_batched_jittered_initial_points():
with pm.Model() as model:
x = pm.MvNormal("x",
mu=np.zeros(3),
cov=np.eye(3),
shape=(2, 3),
initval=np.zeros((2, 3)))
# No jitter
ips = _get_batched_jittered_initial_points(model=model,
chains=1,
random_seed=1,
initvals=None,
jitter=False)
assert np.all(ips[0] == 0)
# Single chain
ips = _get_batched_jittered_initial_points(model=model,
chains=1,
random_seed=1,
initvals=None)
assert ips[0].shape == (2, 3)
assert np.all(ips[0] != 0)
# Multiple chains
ips = _get_batched_jittered_initial_points(model=model,
chains=2,
random_seed=1,
initvals=None)
assert ips[0].shape == (2, 2, 3)
assert np.all(ips[0][0] != ips[0][1])
def mv_prior_simple():
n = 3
noise = 0.1
X = np.linspace(0, 1, n)[:, None]
K = pm.gp.cov.ExpQuad(1, 1)(X).eval()
L = np.linalg.cholesky(K)
K_noise = K + noise * np.eye(n)
obs = floatX_array([-0.1, 0.5, 1.1])
# Posterior mean
L_noise = np.linalg.cholesky(K_noise)
alpha = np.linalg.solve(L_noise.T, np.linalg.solve(L_noise, obs))
mu_post = np.dot(K.T, alpha)
# Posterior standard deviation
v = np.linalg.solve(L_noise, K)
std_post = (K - np.dot(v.T, v)).diagonal()**0.5
with pm.Model() as model:
x = pm.Flat("x", size=n)
x_obs = pm.MvNormal("x_obs", observed=obs, mu=x, cov=noise * np.eye(n))
return model.compute_initial_point(), model, (K, L, mu_post, std_post,
noise)
def _build_prior(self, name, X, reparameterize=True, jitter=JITTER_DEFAULT, **kwargs):
mu = self.mean_func(X)
cov = stabilize(self.cov_func(X), jitter)
if reparameterize:
size = infer_size(X, kwargs.pop("size", None))
v = pm.Normal(name + "_rotated_", mu=0.0, sigma=1.0, size=size, **kwargs)
f = pm.Deterministic(name, mu + cholesky(cov).dot(v))
else:
f = pm.MvNormal(name, mu=mu, cov=cov, **kwargs)
return f
def marginal_likelihood(self, name, X, y, noise, is_observed=True, **kwargs):
R"""
Returns the marginal likelihood distribution, given the input
locations `X` and the data `y`.
This is integral over the product of the GP prior and a normal likelihood.
.. math::
y \mid X,\theta \sim \int p(y \mid f,\, X,\, \theta) \, p(f \mid X,\, \theta) \, df
Parameters
----------
name: string
Name of the random variable
X: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
y: array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise: scalar, Variable, or Covariance
Standard deviation of the Gaussian noise. Can also be a Covariance for
non-white noise.
is_observed: bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
if not isinstance(noise, Covariance):
noise = pm.gp.cov.WhiteNoise(noise)
mu, cov = self._build_marginal_likelihood(X, noise)
self.X = X
self.y = y
self.noise = noise
if is_observed:
return pm.MvNormal(name, mu=mu, cov=cov, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, size=shape, **kwargs)
def test_square():
iw = pymc.Wishart("A", 2, np.eye(2))
mnc = pymc.MvNormal(
"v",
np.zeros(2),
iw,
value=np.zeros(2),
observed=True)
M = pymc.MCMC([iw, mnc])
M.sample(8, progress_bar=0)
def getModel():
C = pm.Categorical('1-Cat', [0.2, 0.4, 0.1, 0.3])
#@UndefinedVariable
# C = pm.Categorical('1-Cat', [0.2, 0.4, 0.1, 0.3], observed=True, value=3); #@UndefinedVariable
p_N = pm.Lambda(
'p_Norm',
lambda n=C: np.select([n == 0, n == 1, n == 2, n == 3],
[[-5, -5], [0, 0], [5, 5], [10, 10]]),
doc='Pr[Norm|Cat]')
N = pm.MvNormal('2-Norm_2D', mu=p_N, tau=np.eye(2, 2))
#@UndefinedVariable
# N = pm.MvNormal('2-Norm', mu=p_N, tau=np.eye(2,2), observed=True, value=[2.5,2.5]); #@UndefinedVariable
return pm.Model([C, N])
def mv_simple():
mu = floatX_array([-0.1, 0.5, 1.1])
p = floatX_array([[2.0, 0, 0], [0.05, 0.1, 0], [1.0, -0.05, 5.5]])
tau = np.dot(p, p.T)
with pm.Model() as model:
pm.MvNormal(
"x",
at.constant(mu),
tau=at.constant(tau),
initval=floatX_array([0.1, 1.0, 0.8]),
)
H = tau
C = np.linalg.inv(H)
return model.compute_initial_point(), model, (mu, C)
def getModel():
D = pm.Dirichlet('1-Dirichlet', theta=[2, 1, 2, 4])
#@UndefinedVariable
# p_B = pm.Lambda('p_Bern', lambda b=B: np.where(b==0, 0.9, 0.1), doc='Pr[Bern|Beta]');
C = pm.Categorical('2-Cat', D)
#@UndefinedVariable
# C = pm.Categorical('1-Cat', [0.2, 0.4, 0.1, 0.3], observed=True, value=3); #@UndefinedVariable
p_N = pm.Lambda(
'p_Norm',
lambda n=C: np.select([n == 0, n == 1, n == 2, n == 3],
[[-5, -5], [0, 0], [5, 5], [10, 10]]),
doc='Pr[Norm|Cat]')
N = pm.MvNormal('3-Norm_2D', mu=p_N, tau=np.eye(2, 2))
#@UndefinedVariable
# N = pm.MvNormal('2-Norm_2D', mu=p_N, tau=np.eye(2,2), observed=True, value=[2.5,2.5]); #@UndefinedVariable
return pm.Model([D, C, N])
def conditional(self,
name,
Xnew,
pred_noise=False,
given=None,
jitter=0.0,
**kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that the GP prior was over, the
conditional distribution over a set of new points, `f_*` is:
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} f \,,
K(X_*, X_*) - K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} K(X, X_*) \right)
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise: bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given: dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC for more information.
jitter: scalar
A small correction added to the diagonal of positive semi-definite
covariance matrices to ensure numerical stability. For conditionals
the default value is 0.0.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens,
jitter)
return pm.MvNormal(name, mu=mu, cov=cov, **kwargs)
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 __init__(self,
predictions,
measurements,
uncertainties,
regularization_strength=1.0,
precision=None,
prior_pops=None):
"""Bayesian Energy Landscape Tilting with MultiVariate Normal Prior.
Parameters
----------
predictions : ndarray, shape = (num_frames, num_measurements)
predictions[j, i] gives the ith observabled predicted at frame j
measurements : ndarray, shape = (num_measurements)
measurements[i] gives the ith experimental measurement
uncertainties : ndarray, shape = (num_measurements)
uncertainties[i] gives the uncertainty of the ith experiment
regularization_strength : float
How strongly to weight the MVN prior (e.g. lambda)
precision : ndarray, optional, shape = (num_measurements, num_measurements)
The precision matrix of the predicted observables.
prior_pops : ndarray, optional, shape = (num_frames)
Prior populations of each conformation. If None, use uniform populations.
"""
BELT.__init__(self,
predictions,
measurements,
uncertainties,
prior_pops=prior_pops)
if precision == None:
precision = np.cov(predictions.T)
if precision.ndim == 0:
precision = precision.reshape((1, 1))
self.alpha = pymc.MvNormal("alpha",
np.zeros(self.num_measurements),
tau=precision * regularization_strength)
self.initialize_variables()
def test_full_adapt_sampling(seed=289586):
np.random.seed(seed)
L = np.random.randn(5, 5)
L[np.diag_indices_from(L)] = np.exp(L[np.diag_indices_from(L)])
L[np.triu_indices_from(L, 1)] = 0.0
with pymc.Model() as model:
pymc.MvNormal("a", mu=np.zeros(len(L)), chol=L, size=len(L))
initial_point = model.recompute_initial_point()
initial_point_size = sum(initial_point[n.name].size
for n in model.value_vars)
pot = quadpotential.QuadPotentialFullAdapt(
initial_point_size, np.zeros(initial_point_size))
step = pymc.NUTS(model=model, potential=pot)
pymc.sample(draws=10,
tune=1000,
random_seed=seed,
step=step,
cores=1,
chains=1)
def conditional(self, name, Xnew, jitter=0.0, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
`Xnew` will be split by columns and fed to the relevant
covariance functions based on their `input_dim`. For example, if
`cov_func1`, `cov_func2`, and `cov_func3` have `input_dim` of 2,
1, and 4, respectively, then `Xnew` must have 7 columns and a
covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
The distribution returned by `conditional` does not have a
Kronecker structure regardless of whether the input points lie
on a full grid. Therefore, `Xnew` does not need to have grid
structure.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
jitter: scalar
A small correction added to the diagonal of positive semi-definite
covariance matrices to ensure numerical stability. For conditionals
the default value is 0.0.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew, jitter)
return pm.MvNormal(name, mu=mu, cov=cov, **kwargs)
xmin = np.min(xy_meas[:, 0]) * 1.1
ymin = np.min(xy_meas[:, 1]) * 1.1
xmax = np.max(xy_meas[:, 0]) * 1.1
ymax = np.max(xy_meas[:, 1]) * 1.1
# x_nodes = np.array([pymc.Uniform('x_nodes_%i'%i, lower=xmin, upper=xmax) for i in range(Nnodes)])
# y_nodes = np.array([pymc.Uniform('y_nodes_%i'%i, lower=ymin, upper=ymax) for i in range(Nnodes)])
xy_nodes = np.empty(Nnodes, dtype=object)
for i in range(Nnodes):
mu = XYnodes_heuristic[i, 0:2]
Sigma = np.empty((2, 2))
Sigma[0, :] = XYnodes_heuristic[i, 2:]
Sigma[1, :] = Sigma[0, ::-1] #symmetric
Tau = np.linalg.inv(Sigma)
xy_nodes[i] = pymc.MvNormal('xy_nodes_%i' % i, mu=mu, tau=Tau)
# @pymc.deterministic(plot=False)
# def xy_points(o=state_origin, d=state_dest, f=frac, x_n = x_nodes, y_n = y_nodes):
# x_n_s = x_n #np.sort(x_n)
# y_n_s = y_n #np.sort(y_n)
# x = (x_n_s[d]-x_n_s[o])*f + x_n_s[o]
# y = (y_n_s[d]-y_n_s[o])*f + y_n_s[o]
# out = np.column_stack([x, y])
# return out
Prows = np.empty(Nnodes, dtype=object)
for i in range(Nnodes):
t = np.ones(Nnodes) * 10
t[i] = 0.5
Prows[i] = pymc.Dirichlet('Dir_%i' % i, theta=t)
root = sqrt(Omega_Lambda + Omega_M * (1 + x)**3 +
(1 - Omega_Lambda - Omega_M) * (1 + x)**2)
result = 1 / root
return result
for i in range(len(xdata)):
z = xdata[i]
D_L = ((2998 / h) * (1 + z)) * quad(Formula, 0, z)[0]
mu_b[i] = 25 + 5 * log10(abs(D_L))
return mu_b
# (We define the error as 2 std)
T = np.linalg.inv(
my_cov) # tau = precision matrix = inverse of covariance matrix
y = pymc.MvNormal('y', mu=y_model, tau=T, observed=True, value=ydata)
# package the full model in a dictionary
model1 = dict(h=h, Omega_Lambda=Omega_Lambda, Omega_M=Omega_M, y=y)
# run the basic MCMC:
S = pymc.MCMC(model1)
S.sample(iter=100000, burn=80000)
# extract the traces and plot the results
pymc_trace_unifo = [
S.trace('h')[:],
S.trace('Omega_Lambda')[:],
S.trace('Omega_M')[:]
]
plot_MCMC_results(xdata, ydata, pymc_trace_unifo)
print()
print("h mean = {:.4f}".format(pymc_trace_unifo[0].mean()))
import pandas as pd
import pymc as pm
d = pd.read_csv("data/gp.csv")
d.shape
D = np.array([ np.abs(xi - d.x) for xi in d.x])
I = (D == 0).astype("double")
with pm.Model() as gp:
nugget = pm.HalfCauchy("nugget", beta=5)
sigma2 = pm.HalfCauchy("sigma2", beta=5)
ls = pm.HalfCauchy("ls", beta=5)
Sigma = I * nugget + sigma2 * np.exp(-0.5 * D**2 * ls**2)
y = pm.MvNormal(
"y",
mu=np.zeros(d.shape[0]),
cov=Sigma, observed=d.y
)
with gp:
post_nuts = pm.sample(
return_inferencedata = True,
chains = 2
)
def __init__(self, F, G, V, W, m_0, C_0, Y_vals = None):
"""
D = DLM(F, G, V, W, m_0, C_0[, Y_vals])
Returns special NormalSubmodel instance representing the dynamic
linear model formed by F, G, V and W.
Resulting probability model:
theta[0] | m_0, C_0 ~ N(m_0, C_0)
theta[t] | theta[t-1], G[t], W[t] ~ N(G[t] theta[t-1], W[t]), t = 1..T
Y[t] | theta[t], F[t], V[t] ~ N(F[t] theta[t], V[t]), t = 0..T
Arguments F, G, V should be dictionaries keyed by name of component.
F[comp], G[comp], V[comp] should be lists.
F[comp][t] should be the design vector of component 'comp' at time t.
G[comp][t] should be the system matrix.
Argument W should be either a number between 0 and 1 or a dictionary of lists
like V.
If a dictionary of lists, W[comp][t] should be the system covariance or
variance at time t.
If a scalar, W should be the discount factor for the DLM.
Arguments V and Y_vals, if given, should be lists.
V[t] should be the observation covariance or variance at time t.
Y_vals[t] should give the value of output Y at time t.
Arguments m_0 and C_0 should be dictionaries keyed by name of component.
m_0[comp] should be the mean of theta[comp][0].
C_0[comp] should be the covariance or variance of theta[comp][0].
Note: if multiple components are correlated in W or V, they should be made into
a single component.
D.comp is a handle to a list.
D.comp[t] is a Stochastic representing the value of system state 'theta'
sliced according to component 'comp' at time t.
D.theta is a dictionary of lists analogous to F, G, V and W.
D.Y is a list. D.Y[t] is a Stochastic representing the value of the output
'Y' at time t.
"""
self.comps = F.keys()
self.F = dict_to_recarray(F)
self.G = dict_to_recarray(G)
self.V = pymc.ListContainer(V)
if np.isscalar(W):
self.discount = True
self.delta = W
else:
self.W = dict_to_recarray(W)
self.discount = False
self.delta = None
if self.discount:
raise NotImplemented, "Have yet to code up the discount factor."
self.m_0 = dict_to_recarray(m_0)
self.C_0 = dict_to_recarray(C_0)
self.T = len(self.V)
theta = {}
theta_mean = {}
Y_mean = []
Y = []
# ==============
# = Make theta =
# ==============
for comp in self.comps:
# Is diagonal the covariance or variance?
if isinstance(self.W[comp][0], pymc.Variable):
diag = isvector(self.W[comp][0].value)
else:
diag = isvector(self.W[comp][0])
if diag:
# Normal variates if diagonal.
theta[comp] = [pymc.Normal('%s[0]'%comp, m_0[comp], C_0[comp])]
else:
# MV normal otherwise.
theta[comp] = [pymc.MvNormal('%s[0]'%comp, m_0[comp], C_0[comp])]
theta_mean[comp] = []
for t in xrange(1,self.T):
theta_mean[comp].append(pymc.LinearCombination('%s_mean[%i]'%(comp, t), [G[comp][t-1]], [theta[comp][t-1]]))
if diag:
# Normal variates if diagonal.
theta[comp].append(pymc.Normal('%s[%i]'%(comp,t), theta_mean[comp][t-1], W[comp][t-1]))
else:
# MV normal otherwise.
theta[comp].append(pymc.MvNormal('%s[%i]'%(comp,t), theta_mean[comp][t-1], W[comp][t-1]))
self.theta = dict_to_recarray(theta)
self.theta_mean = dict_to_recarray(theta_mean)
# ==========
# = Make Y =
# ==========
Y_diag = isvector(self.V.value[0])
for t in xrange(self.T):
x_coef = []
y_coef = []
for comp in self.comps:
x_coef.append(self.F[comp][t])
y_coef.append(theta[comp][t])
Y_mean.append(pymc.LinearCombination('Y_mean[%i]'%t, x_coef, y_coef))
if Y_diag:
# Normal variates if diagonal.
Y.append(pymc.Normal('Y[%i]'%t, Y_mean[t], V[t]))
else:
# MV normal otherwise.
Y.append(pymc.MvNormal('Y[%i]'%t, Y_mean[t], V[t]))
# If data provided, use it.
if Y_vals is not None:
Y[t].value = Y_vals[t]
Y[t].observed = True
self.Y_mean = pymc.Container(np.array(Y_mean))
self.Y = pymc.Container(np.array(Y))
# No sense creating a NormalSubmodel here... just stay a ListContainer.
NormalSubmodel.__init__(self, [F,G,W,V,m_0,C_0,Y,theta,theta_mean,Y_mean])
ans[i] = counties_dict[counties[i]]
return ans
index_c = createCountyIndex(counties)
# Priors
sigma_y = pymc.Uniform('sigma_y', lower=0, upper=100)
tau_y = pymc.Lambda('tau_y', lambda s=sigma_y: s**-2)
xi = pymc.Uniform('xi', lower=0, upper=100, value=np.zeros(K))
mu_raw = pymc.Normal('mu_raw', mu=0., tau=0.0001,value=np.zeros(K))
Tau_B_raw = pymc.Wishart('Tau_B_raw', df, Tau=np.diag(np.ones(K)))
B_raw_m = np.ones( (J,K) )
B_raw = []
for i in range(J):
B_raw.append(pymc.MvNormal('B_raw_%i' % i, mu_raw, Tau_B_raw))
@pymc.deterministic
def Sigma_B_raw(Tau_B_raw=Tau_B_raw):
return np.linalg.inv(Tau_B_raw)
@pymc.deterministic
def rho_B(Sigma_B_raw=Sigma_B_raw):
return Sigma_B_raw / np.sqrt(np.diag(Sigma_B_raw) * Sigma_B_raw)
@pymc.deterministic
def Sigma_B(xi=xi,Sigma_B_raw=Sigma_B_raw):
return abs(xi) * np.sqrt(np.diag(Sigma_B_raw))
@pymc.deterministic(plot=True)
def B(xi=xi, B_raw=B_raw, B_raw_m=B_raw_m):
import pymc
import numpy as np
import logging
log = logging.getLogger('emissions')
log_2_pi = np.log(2 * np.pi)
invwishart = lambda nu, L: pymc.InverseWishart("invwishart", nu, L).random()
mvnormal = lambda mu, tau: pymc.MvNormal('mvnormal', mu, tau).random()
#invwishart_like = lambda x, nu, L: pymc.inverse_wishart_like(x,nu,L)
class Gaussian:
def __init__(self, nu, Lambda, mu_0, kappa, mu, tau):
self.nu = nu
self.Lambda = Lambda
self.mu_0 = mu_0 # mu_0 is the mean of the prior on the mean
self.kappa = float(kappa)
self.mu = mu # mu is the current value of the mean for each state
self.tau = tau # tau is the current precision matrix for each state
self.states = range(len(mu))
self.K = len(self.states)
def likelihood(self, state, obs):
assert state in self.states, (state, self.states)
return pymc.mv_normal_like(obs, self.mu[state], self.tau[state])
def sample_obs(self, state):
assert state in self.states, (state, self.states)
import utils
import pymc
from params import * # noqa
if __name__ == '__main__':
# load a dataset
dataset = utils.load_dataset(DATASET_PATH)
observed_xs = dataset[:, 0]
observed_ys = dataset[:, 1]
# plot_ground_truth()
# define a prior for w, which is a multivariable gaussian
ws = pymc.MvNormal('ws', mu=np.zeros(M), tau=ALPHA * np.eye(M), value=np.zeros(M))
# calculate x^0, x^1, ..., x^{M-1}
xs = np.empty((M, len(observed_xs)), dtype=np.float32)
for i in range(M):
xs[i] = np.power(observed_xs, i)
assert(xs.shape == (M, len(observed_xs)))
# define a deterministic function
linear_regression = pymc.Lambda('linear_regression', lambda xs=xs, ws=ws: ws.dot(xs))
# define a model likelihood
y = pymc.Normal('y', mu=linear_regression, tau=TAU, value=observed_ys, observed=True)
# make a model
model = pymc.Model([y, ws])
# priors, fairly vague
prior_mean = data.mean(0)
sigma0 = np.diag([1., 1.])
prior_cov = np.cov(data.T)
# shared hyperparameter?
# theta_tau = pm.Wishart('theta_tau', n=4, Tau=L.inv(sigma0))
# df = pm.DiscreteUniform('df', 3, 50)
thetas = []
taus = []
for j in range(ncomps):
# need a hyperparameter for degrees of freedom?
tau = pm.Wishart('C_%d' % j, n=3, Tau=inv(prior_cov))
theta = pm.MvNormal('theta_%d' % j, mu=prior_mean, tau=inv(2 * prior_cov))
thetas.append(theta)
taus.append(tau)
alpha0 = np.ones(3.) / 3
weights = pm.Dirichlet('weights', theta=alpha0)
# labels = pm.Categorical('labels', p=weights, size=len(data))
from pandas.util.testing import set_trace as st
import pdfs
import util
def mixture_loglike(data, thetas, covs, labels):
