def run(n_samples=3000):
model = build_model()
start = model.test_point
h = pm.find_hessian(start, model=model)
step = pm.Metropolis(model.vars, h, blocked=True, model=model)
trace = pm.sample(n_samples, step, start, model=model)
return trace
def run(n_samples=3000):
model = build_model()
start = model.test_point
h = pm.find_hessian(start, model=model)
step = pm.Metropolis(model.vars, h, blocked=True, model=model)
trace = pm.sample(n_samples, step=step, start=start, model=model)
return trace
def init_nuts(init='advi', n_init=500000, model=None, **kwargs):
"""Initialize and sample from posterior of a continuous model.
This is a convenience function. NUTS convergence and sampling speed is extremely
dependent on the choice of mass/scaling matrix. In our experience, using ADVI
to estimate a diagonal covariance matrix and using this as the scaling matrix
produces robust results over a wide class of continuous models.
Parameters
----------
init : str {'advi', 'advi_map', 'map', 'nuts'}
Initialization method to use.
* advi : Run ADVI to estimate posterior mean and diagonal covariance matrix.
* advi_map: Initialize ADVI with MAP and use MAP as starting point.
* map : Use the MAP as starting point.
* nuts : Run NUTS and estimate posterior mean and covariance matrix.
n_init : int
Number of iterations of initializer
If 'advi', number of iterations, if 'metropolis', number of draws.
model : Model (optional if in `with` context)
**kwargs : keyword arguments
Extra keyword arguments are forwarded to pymc3.NUTS.
Returns
-------
start, nuts_sampler
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
pm._log.info('Initializing NUTS using {}...'.format(init))
if init == 'advi':
v_params = pm.variational.advi(n=n_init)
start = pm.variational.sample_vp(v_params, 1, progressbar=False)[0]
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'advi_map':
start = pm.find_MAP()
v_params = pm.variational.advi(n=n_init, start=start)
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'map':
start = pm.find_MAP()
cov = pm.find_hessian(point=start)
elif init == 'nuts':
init_trace = pm.sample(step=pm.NUTS(), draws=n_init)
cov = pm.trace_cov(init_trace[n_init//2:])
start = {varname: np.mean(init_trace[varname]) for varname in init_trace.varnames}
else:
raise NotImplemented('Initializer {} is not supported.'.format(init))
step = pm.NUTS(scaling=cov, is_cov=True, **kwargs)
return start, step
def _laplace(model):
"""
Fit a model using a laplace approximation. Mainly for pedagogical use. ``mcmc`` and ``advi``
are better approximations
Parameters
----------
model: PyMC3 model
Returns
-------
Dictionary, the keys are the names of the variables and the values tuples of modes and standard
deviations.
"""
with model:
varis = [v for v in model.unobserved_RVs if not pm.util.is_transformed_name(v.name)]
maps = pm.find_MAP(start=model.test_point, vars=varis)
hessian = pm.find_hessian(maps, vars=varis)
if np.linalg.det(hessian) == 0:
raise np.linalg.LinAlgError("Singular matrix. Use mcmc or advi method")
stds = np.diag(np.linalg.inv(hessian) ** 0.5)
maps = [v for (k, v) in maps.items() if not pm.util.is_transformed_name(k)]
modes = [v.item() if v.size == 1 else v for v in maps]
names = [v.name for v in varis]
shapes = [np.atleast_1d(mode).shape for mode in modes]
stds_reshaped = []
idx0 = 0
for shape in shapes:
idx1 = idx0 + sum(shape)
stds_reshaped.append(np.reshape(stds[idx0:idx1], shape))
idx0 = idx1
return dict(zip(names, zip(modes, stds_reshaped)))
def test_plots_multidimensional():
# Test single trace
from .models import multidimensional_model
start, model, _ = multidimensional_model()
with model as model:
h = np.diag(find_hessian(start))
step = Metropolis(model.vars, h)
trace = sample(3000, step, start)
traceplot(trace)
def test_plots_multidimensional():
# Test single trace
from .models import multidimensional_model
start, model, _ = multidimensional_model()
with model as model:
h = np.diag(find_hessian(start))
step = Metropolis(model.vars, h)
trace = sample(3000, step, start)
traceplot(trace)
def test_plots():
# Test single trace
from pymc3.examples import arbitrary_stochastic as asmod
with asmod.model as model:
start = model.test_point
h = find_hessian(start)
step = Metropolis(model.vars, h)
trace = sample(3000, step, start)
traceplot(trace)
forestplot(trace)
autocorrplot(trace)
def test_plots():
# Test single trace
from pymc3.examples import arbitrary_stochastic as asmod
with asmod.model as model:
start = model.test_point
h = find_hessian(start)
step = Metropolis(model.vars, h)
trace = sample(3000, step, start)
traceplot(trace)
forestplot(trace)
autocorrplot(trace)
def make_normal_approx(vars):
'''
Gets the normal approximation for the posterior near its maximum for the specified variables.
Useful for getting quick summaries without sampling from, e.g. linear models.
Don't use for complex models!
Parameters:
vars: a list of variable names
Returns:
a DataFrame with the MAP estimates, standard deviations, and compatible intervals estimated from the Hessian
'''
map_est = pm.find_MAP()
std_est = (1/pm.find_hessian(map_est, vars=vars))**0.5
data = []
for var in vars:
i = vars.index(var)
cv = sp.stats.norm.ppf(0.97)
data.append([map_est[var.name].round(3), std_est[i,i].round(3), (map_est[var.name] - std_est[i, i] * cv).round(3), (map_est[var.name] + std_est[i, i] * cv).round(3)])
summary = pd.DataFrame(data, columns = ['map', 'sd', 'hdi_3%', 'hdi_97%'], index = vars)
return summary
def test_leapfrog_reversible():
n = 3
start, model, _ = models.non_normal(n)
with model:
h = pm.find_hessian(start, model=model)
step = pm.HamiltonianMC(model.vars, h, model=model)
bij = DictToArrayBijection(step.ordering, start)
logp, dlogp = list(map(bij.mapf, step.fs))
H = Hamiltonian(logp, dlogp, step.potential)
q0 = bij.map(start)
p0 = np.ones(n)*.05
for e in [.01, .1, 1.2]:
for L in [1, 2, 3, 4, 20]:
q, p = q0, p0
q, p = leapfrog(H, q, p, L, e)
q, p = leapfrog(H, q, -p, L, e)
close_to(q, q0, 1e-8, str((L, e)))
close_to(-p, p0, 1e-8, str((L, e)))
def test_leapfrog_reversible():
n = 3
start, model, _ = models.non_normal(n)
with model:
h = pm.find_hessian(start, model=model)
step = pm.HamiltonianMC(model.vars, h, model=model)
bij = pm.DictToArrayBijection(step.ordering, start)
logp, dlogp = list(map(bij.mapf, step.fs))
H = Hamiltonian(logp, dlogp, step.potential)
q0 = bij.map(start)
p0 = np.ones(n)*.05
for e in [.01, .1, 1.2]:
for L in [1, 2, 3, 4, 20]:
q, p = q0, p0
q, p = leapfrog(H, q, p, L, e)
q, p = leapfrog(H, q, -p, L, e)
close_to(q, q0, 1e-8, str((L, e)))
close_to(-p, p0, 1e-8, str((L, e)))
# model equation
sex_idx = data.sex.values
height_mu = intercept[sex_idx] + beta[sex_idx] * data.weight
mc.Normal('height', mu=height_mu, sd=error, observed=data.height)
# In[78]:
model.vars
# In[79]:
with model:
start = mc.find_MAP()
step = mc.NUTS(state=start)
hessian = mc.find_hessian(start)
trace = mc.sample(5000, step, start=start)
# In[80]:
fig, axes = plt.subplots(3, 2, figsize=(8, 6), squeeze=False)
mc.traceplot(trace, vars=['intercept', 'beta', 'error'], ax=axes)
fig.tight_layout()
fig.savefig("ch16-multilevel-sample-trace.pdf")
fig.savefig("ch16-multilevel-sample-trace.png")
# In[81]:
intercept_m, intercept_f = trace.get_values('intercept').mean(axis=0)
# In[82]:
import pymc3 as pm
with pm.Model() as model:
x = pm.Normal('x', 1, 1)
x2 = pm.Potential('x2', -x**2)
start = model.test_point
h = pm.find_hessian(start)
step = pm.Metropolis(model.vars, h)
def run(n=3000):
if n == "short":
n = 50
with model:
trace = pm.sample(n, step=step, start=start)
if __name__ == '__main__':
run()
def init_nuts(init='auto', njobs=1, n_init=500000, model=None,
random_seed=-1, progressbar=True, **kwargs):
"""Set up the mass matrix initialization for NUTS.
NUTS convergence and sampling speed is extremely dependent on the
choice of mass/scaling matrix. This function implements different
methods for choosing or adapting the mass matrix.
Parameters
----------
init : str
Initialization method to use.
* auto : Choose a default initialization method automatically.
Currently, this is `'jitter+adapt_diag'`, but this can change in
the future. If you depend on the exact behaviour, choose an
initialization method explicitly.
* adapt_diag : Start with a identity mass matrix and then adapt
a diagonal based on the variance of the tuning samples. All
chains use the test value (usually the prior mean) as starting
point.
* jitter+adapt_diag : Same as `adapt_diag`, but add uniform jitter
in [-1, 1] to the starting point in each chain.
* advi+adapt_diag : Run ADVI and then adapt the resulting diagonal
mass matrix based on the sample variance of the tuning samples.
* advi+adapt_diag_grad : Run ADVI and then adapt the resulting
diagonal mass matrix based on the variance of the gradients
during tuning. This is **experimental** and might be removed
in a future release.
* advi : Run ADVI to estimate posterior mean and diagonal mass
matrix.
* advi_map: Initialize ADVI with MAP and use MAP as starting point.
* map : Use the MAP as starting point. This is discouraged.
* nuts : Run NUTS and estimate posterior mean and mass matrix from
the trace.
njobs : int
Number of parallel jobs to start.
n_init : int
Number of iterations of initializer
If 'ADVI', number of iterations, if 'nuts', number of draws.
model : Model (optional if in `with` context)
progressbar : bool
Whether or not to display a progressbar for advi sampling.
**kwargs : keyword arguments
Extra keyword arguments are forwarded to pymc3.NUTS.
Returns
-------
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
vars = kwargs.get('vars', model.vars)
if set(vars) != set(model.vars):
raise ValueError('Must use init_nuts on all variables of a model.')
if not pm.model.all_continuous(vars):
raise ValueError('init_nuts can only be used for models with only '
'continuous variables.')
if not isinstance(init, str):
raise TypeError('init must be a string.')
if init is not None:
init = init.lower()
if init == 'auto':
init = 'jitter+adapt_diag'
pm._log.info('Initializing NUTS using {}...'.format(init))
random_seed = int(np.atleast_1d(random_seed)[0])
cb = [
pm.callbacks.CheckParametersConvergence(
tolerance=1e-2, diff='absolute'),
pm.callbacks.CheckParametersConvergence(
tolerance=1e-2, diff='relative'),
]
if init == 'adapt_diag':
start = [model.test_point] * njobs
mean = np.mean([model.dict_to_array(vals) for vals in start], axis=0)
var = np.ones_like(mean)
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, var, 10)
if njobs == 1:
start = start[0]
elif init == 'jitter+adapt_diag':
start = []
for _ in range(njobs):
mean = {var: val.copy() for var, val in model.test_point.items()}
for val in mean.values():
val[...] += 2 * np.random.rand(*val.shape) - 1
start.append(mean)
mean = np.mean([model.dict_to_array(vals) for vals in start], axis=0)
var = np.ones_like(mean)
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, var, 10)
if njobs == 1:
start = start[0]
elif init == 'advi+adapt_diag_grad':
approx = pm.fit(
random_seed=random_seed,
n=n_init, method='advi', model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window,
) # type: pm.MeanField
start = approx.sample(draws=njobs)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds) ** 2
mean = approx.bij.rmap(approx.mean.get_value())
mean = model.dict_to_array(mean)
weight = 50
potential = quadpotential.QuadPotentialDiagAdaptGrad(
model.ndim, mean, cov, weight)
if njobs == 1:
start = start[0]
elif init == 'advi+adapt_diag':
approx = pm.fit(
random_seed=random_seed,
n=n_init, method='advi', model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window,
) # type: pm.MeanField
start = approx.sample(draws=njobs)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds) ** 2
mean = approx.bij.rmap(approx.mean.get_value())
mean = model.dict_to_array(mean)
weight = 50
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, cov, weight)
if njobs == 1:
start = start[0]
elif init == 'advi':
approx = pm.fit(
random_seed=random_seed,
n=n_init, method='advi', model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window
) # type: pm.MeanField
start = approx.sample(draws=njobs)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds) ** 2
potential = quadpotential.QuadPotentialDiag(cov)
if njobs == 1:
start = start[0]
elif init == 'advi_map':
start = pm.find_MAP()
approx = pm.MeanField(model=model, start=start)
pm.fit(
random_seed=random_seed,
n=n_init, method=pm.KLqp(approx),
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window
)
start = approx.sample(draws=njobs)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds) ** 2
potential = quadpotential.QuadPotentialDiag(cov)
if njobs == 1:
start = start[0]
elif init == 'map':
start = pm.find_MAP()
cov = pm.find_hessian(point=start)
start = [start] * njobs
potential = quadpotential.QuadPotentialFull(cov)
if njobs == 1:
start = start[0]
elif init == 'nuts':
init_trace = pm.sample(draws=n_init, step=pm.NUTS(),
tune=n_init // 2,
random_seed=random_seed)
cov = np.atleast_1d(pm.trace_cov(init_trace))
start = list(np.random.choice(init_trace, njobs))
potential = quadpotential.QuadPotentialFull(cov)
if njobs == 1:
start = start[0]
else:
raise NotImplementedError('Initializer {} is not supported.'.format(init))
step = pm.NUTS(potential=potential, **kwargs)
return start, step
import pymc3 as pm
with pm.Model() as model:
x = pm.Normal('x', 1, 1)
x2 = pm.Potential('x2', -x ** 2)
start = model.test_point
h = pm.find_hessian(start)
step = pm.Metropolis(model.vars, h)
def run(n=3000):
if n == "short":
n = 50
with model:
pm.sample(n, step=step, start=start)
if __name__ == '__main__':
run()
def init_nuts(init='ADVI',
njobs=1,
n_init=500000,
model=None,
random_seed=-1,
progressbar=True,
**kwargs):
"""Initialize and sample from posterior of a continuous model.
This is a convenience function. NUTS convergence and sampling speed is extremely
dependent on the choice of mass/scaling matrix. In our experience, using ADVI
to estimate a diagonal covariance matrix and using this as the scaling matrix
produces robust results over a wide class of continuous models.
Parameters
----------
init : str {'ADVI', 'ADVI_MAP', 'MAP', 'NUTS'}
Initialization method to use.
* ADVI : Run ADVI to estimate posterior mean and diagonal covariance matrix.
* ADVI_MAP: Initialize ADVI with MAP and use MAP as starting point.
* MAP : Use the MAP as starting point.
* NUTS : Run NUTS and estimate posterior mean and covariance matrix.
njobs : int
Number of parallel jobs to start.
n_init : int
Number of iterations of initializer
If 'ADVI', number of iterations, if 'metropolis', number of draws.
model : Model (optional if in `with` context)
progressbar : bool
Whether or not to display a progressbar for advi sampling.
**kwargs : keyword arguments
Extra keyword arguments are forwarded to pymc3.NUTS.
Returns
-------
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
pm._log.info('Initializing NUTS using {}...'.format(init))
random_seed = int(np.atleast_1d(random_seed)[0])
if init is not None:
init = init.lower()
cb = [
pm.callbacks.CheckParametersConvergence(tolerance=1e-2,
diff='absolute'),
pm.callbacks.CheckParametersConvergence(tolerance=1e-2,
diff='relative'),
]
if init == 'advi':
approx = pm.fit(random_seed=random_seed,
n=n_init,
method='advi',
model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window) # type: pm.MeanField
start = approx.sample(draws=njobs)
stds = approx.gbij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
if njobs == 1:
start = start[0]
elif init == 'advi_map':
start = pm.find_MAP()
approx = pm.MeanField(model=model, start=start)
pm.fit(random_seed=random_seed,
n=n_init,
method=pm.ADVI.from_mean_field(approx),
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window)
start = approx.sample(draws=njobs)
stds = approx.gbij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
if njobs == 1:
start = start[0]
elif init == 'map':
start = pm.find_MAP()
cov = pm.find_hessian(point=start)
elif init == 'nuts':
init_trace = pm.sample(draws=n_init,
step=pm.NUTS(),
tune=n_init // 2,
random_seed=random_seed)
cov = np.atleast_1d(pm.trace_cov(init_trace))
start = np.random.choice(init_trace, njobs)
if njobs == 1:
start = start[0]
else:
raise NotImplementedError(
'Initializer {} is not supported.'.format(init))
step = pm.NUTS(scaling=cov, is_cov=True, **kwargs)
return start, step
def init_nuts(init='ADVI',
njobs=1,
n_init=500000,
model=None,
random_seed=-1,
**kwargs):
"""Initialize and sample from posterior of a continuous model.
This is a convenience function. NUTS convergence and sampling speed is extremely
dependent on the choice of mass/scaling matrix. In our experience, using ADVI
to estimate a diagonal covariance matrix and using this as the scaling matrix
produces robust results over a wide class of continuous models.
Parameters
----------
init : str {'ADVI', 'ADVI_MAP', 'MAP', 'NUTS'}
Initialization method to use.
* ADVI : Run ADVI to estimate posterior mean and diagonal covariance matrix.
* ADVI_MAP: Initialize ADVI with MAP and use MAP as starting point.
* MAP : Use the MAP as starting point.
* NUTS : Run NUTS and estimate posterior mean and covariance matrix.
njobs : int
Number of parallel jobs to start.
n_init : int
Number of iterations of initializer
If 'ADVI', number of iterations, if 'metropolis', number of draws.
model : Model (optional if in `with` context)
**kwargs : keyword arguments
Extra keyword arguments are forwarded to pymc3.NUTS.
Returns
-------
start, nuts_sampler
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
pm._log.info('Initializing NUTS using {}...'.format(init))
random_seed = int(np.atleast_1d(random_seed)[0])
if init is not None:
init = init.lower()
if init == 'advi':
v_params = pm.variational.advi(n=n_init, random_seed=random_seed)
start = pm.variational.sample_vp(v_params,
njobs,
progressbar=False,
hide_transformed=False,
random_seed=random_seed)
if njobs == 1:
start = start[0]
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'advi_map':
start = pm.find_MAP()
v_params = pm.variational.advi(n=n_init,
start=start,
random_seed=random_seed)
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'map':
start = pm.find_MAP()
cov = pm.find_hessian(point=start)
elif init == 'nuts':
init_trace = pm.sample(step=pm.NUTS(),
draws=n_init,
random_seed=random_seed)[n_init // 2:]
cov = np.atleast_1d(pm.trace_cov(init_trace))
start = np.random.choice(init_trace, njobs)
if njobs == 1:
start = start[0]
else:
raise NotImplementedError(
'Initializer {} is not supported.'.format(init))
step = pm.NUTS(scaling=cov, is_cov=True, **kwargs)
return start, step
sf = dx_grid / dx_exact # Jacobian scale factor
plt.figure()
#plt.stem(grid, posterior, use_line_collection=True)
plt.bar(grid, posterior, width=1 / n, alpha=0.2)
plt.plot(xs, post_exact * sf)
plt.title('grid approximation')
plt.yticks([])
plt.xlabel('θ')
plt.savefig('../figures/bb_grid.pdf')
# Laplace
with pm.Model() as normal_aproximation:
theta = pm.Beta('theta', 1., 1.)
y = pm.Binomial('y', n=1, p=theta, observed=data) # Bernoulli
mean_q = pm.find_MAP()
std_q = ((1 / pm.find_hessian(mean_q, vars=[theta]))**0.5)[0]
mu = mean_q['theta']
print([mu, std_q])
plt.figure()
plt.plot(xs, stats.norm.pdf(xs, mu, std_q), '--', label='Laplace')
post_exact = stats.beta.pdf(xs, h + 1, t + 1)
plt.plot(xs, post_exact, label='exact')
plt.title('Quadratic approximation')
plt.xlabel('θ', fontsize=14)
plt.yticks([])
plt.legend()
plt.savefig('../figures/bb_laplace.pdf')
# HMC
def init_nuts(init='auto',
chains=1,
n_init=500000,
model=None,
random_seed=None,
progressbar=True,
**kwargs):
"""Set up the mass matrix initialization for NUTS.
NUTS convergence and sampling speed is extremely dependent on the
choice of mass/scaling matrix. This function implements different
methods for choosing or adapting the mass matrix.
Parameters
----------
init : str
Initialization method to use.
* auto : Choose a default initialization method automatically.
Currently, this is `'jitter+adapt_diag'`, but this can change in
the future. If you depend on the exact behaviour, choose an
initialization method explicitly.
* adapt_diag : Start with a identity mass matrix and then adapt
a diagonal based on the variance of the tuning samples. All
chains use the test value (usually the prior mean) as starting
point.
* jitter+adapt_diag : Same as `adapt_diag`, but add uniform jitter
in [-1, 1] to the starting point in each chain.
* advi+adapt_diag : Run ADVI and then adapt the resulting diagonal
mass matrix based on the sample variance of the tuning samples.
* advi+adapt_diag_grad : Run ADVI and then adapt the resulting
diagonal mass matrix based on the variance of the gradients
during tuning. This is **experimental** and might be removed
in a future release.
* advi : Run ADVI to estimate posterior mean and diagonal mass
matrix.
* advi_map: Initialize ADVI with MAP and use MAP as starting point.
* map : Use the MAP as starting point. This is discouraged.
* nuts : Run NUTS and estimate posterior mean and mass matrix from
the trace.
chains : int
Number of jobs to start.
n_init : int
Number of iterations of initializer
If 'ADVI', number of iterations, if 'nuts', number of draws.
model : Model (optional if in `with` context)
progressbar : bool
Whether or not to display a progressbar for advi sampling.
**kwargs : keyword arguments
Extra keyword arguments are forwarded to pymc3.NUTS.
Returns
-------
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
vars = kwargs.get('vars', model.vars)
if set(vars) != set(model.vars):
raise ValueError('Must use init_nuts on all variables of a model.')
if not pm.model.all_continuous(vars):
raise ValueError('init_nuts can only be used for models with only '
'continuous variables.')
if not isinstance(init, str):
raise TypeError('init must be a string.')
if init is not None:
init = init.lower()
if init == 'auto':
init = 'jitter+adapt_diag'
pm._log.info('Initializing NUTS using {}...'.format(init))
if random_seed is not None:
random_seed = int(np.atleast_1d(random_seed)[0])
np.random.seed(random_seed)
cb = [
pm.callbacks.CheckParametersConvergence(tolerance=1e-2,
diff='absolute'),
pm.callbacks.CheckParametersConvergence(tolerance=1e-2,
diff='relative'),
]
if init == 'adapt_diag':
start = [model.test_point] * chains
mean = np.mean([model.dict_to_array(vals) for vals in start], axis=0)
var = np.ones_like(mean)
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, var, 10)
elif init == 'jitter+adapt_diag':
start = []
for _ in range(chains):
mean = {var: val.copy() for var, val in model.test_point.items()}
for val in mean.values():
val[...] += 2 * np.random.rand(*val.shape) - 1
start.append(mean)
mean = np.mean([model.dict_to_array(vals) for vals in start], axis=0)
var = np.ones_like(mean)
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, var, 10)
elif init == 'advi+adapt_diag_grad':
approx = pm.fit(
random_seed=random_seed,
n=n_init,
method='advi',
model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window,
) # type: pm.MeanField
start = approx.sample(draws=chains)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
mean = approx.bij.rmap(approx.mean.get_value())
mean = model.dict_to_array(mean)
weight = 50
potential = quadpotential.QuadPotentialDiagAdaptGrad(
model.ndim, mean, cov, weight)
elif init == 'advi+adapt_diag':
approx = pm.fit(
random_seed=random_seed,
n=n_init,
method='advi',
model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window,
) # type: pm.MeanField
start = approx.sample(draws=chains)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
mean = approx.bij.rmap(approx.mean.get_value())
mean = model.dict_to_array(mean)
weight = 50
potential = quadpotential.QuadPotentialDiagAdapt(
model.ndim, mean, cov, weight)
elif init == 'advi':
approx = pm.fit(random_seed=random_seed,
n=n_init,
method='advi',
model=model,
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window) # type: pm.MeanField
start = approx.sample(draws=chains)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
potential = quadpotential.QuadPotentialDiag(cov)
elif init == 'advi_map':
start = pm.find_MAP(include_transformed=True)
approx = pm.MeanField(model=model, start=start)
pm.fit(random_seed=random_seed,
n=n_init,
method=pm.KLqp(approx),
callbacks=cb,
progressbar=progressbar,
obj_optimizer=pm.adagrad_window)
start = approx.sample(draws=chains)
start = list(start)
stds = approx.bij.rmap(approx.std.eval())
cov = model.dict_to_array(stds)**2
potential = quadpotential.QuadPotentialDiag(cov)
elif init == 'map':
start = pm.find_MAP(include_transformed=True)
cov = pm.find_hessian(point=start)
start = [start] * chains
potential = quadpotential.QuadPotentialFull(cov)
elif init == 'nuts':
init_trace = pm.sample(draws=n_init,
step=pm.NUTS(),
tune=n_init // 2,
random_seed=random_seed)
cov = np.atleast_1d(pm.trace_cov(init_trace))
start = list(np.random.choice(init_trace, chains))
potential = quadpotential.QuadPotentialFull(cov)
else:
raise NotImplementedError(
'Initializer {} is not supported.'.format(init))
step = pm.NUTS(potential=potential, **kwargs)
return start, step
def init_nuts(init='advi', n_init=500000, model=None):
"""Initialize and sample from posterior of a continuous model.
This is a convenience function. NUTS convergence and sampling speed is extremely
dependent on the choice of mass/scaling matrix. In our experience, using ADVI
to estimate a diagonal covariance matrix and using this as the scaling matrix
produces robust results over a wide class of continuous models.
Parameters
----------
init : str {'advi', 'advi_map', 'map', 'nuts'}
Initialization method to use.
* advi : Run ADVI to estimate posterior mean and diagonal covariance matrix.
* advi_map: Initialize ADVI with MAP and use MAP as starting point.
* map : Use the MAP as starting point.
* nuts : Run NUTS and estimate posterior mean and covariance matrix.
n_init : int
Number of iterations of initializer
If 'advi', number of iterations, if 'metropolis', number of draws.
model : Model (optional if in `with` context)
Returns
-------
start, nuts_sampler
start : pymc3.model.Point
Starting point for sampler
nuts_sampler : pymc3.step_methods.NUTS
Instantiated and initialized NUTS sampler object
"""
model = pm.modelcontext(model)
pm._log.info('Initializing NUTS using {}...'.format(init))
if init == 'advi':
v_params = pm.variational.advi(n=n_init)
start = pm.variational.sample_vp(v_params, 1, progressbar=False)[0]
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'advi_map':
start = pm.find_MAP()
v_params = pm.variational.advi(n=n_init, start=start)
cov = np.power(model.dict_to_array(v_params.stds), 2)
elif init == 'map':
start = pm.find_MAP()
cov = pm.find_hessian(point=start)
elif init == 'nuts':
init_trace = pm.sample(step=pm.NUTS(), draws=n_init)
cov = pm.trace_cov(init_trace[n_init // 2:])
start = {
varname: np.mean(init_trace[varname])
for varname in init_trace.varnames
}
else:
raise NotImplemented('Initializer {} is not supported.'.format(init))
step = pm.NUTS(scaling=cov, is_cov=True)
return start, step
_m.x = np.linspace(-0.1, 1.1, 100)
_m.f_x = 2 * _m.x**2 * (1 - _m.x)**2
plt.plot(_m.x, _m.f_x)
# As the analytical solution shows we have three points where the derivative is zero: $x=0$, $x=1$, and $x=\frac{1}{2}$. The value $x=\frac{1}{2}$ maximizes the function, and is the answer.
# # 2.6
# +
_26 = Object()
_26.data = np.repeat((0, 1), (3, 6))
with pm.Model() as _26.na:
_26.p = pm.Uniform('p', 0, 1)
_26.w = pm.Binomial('w', n=len(_26.data), p=_26.p, observed=_26.data.sum())
_26.mean_p = pm.find_MAP()
_26.std_q = ((1 / pm.find_hessian(_26.mean_p, vars=[_26.p]))**0.5)[0]
_26.mean_p['p'], _26.std_q
# -
# Assuming the posterior is Gaussian, it's maximized at $0.67$ and its standard deviation is $0.16$.
# 89% confidence interval:
_26.norm_dist = stats.norm(_26.mean_p['p'], _26.std_q)
_26.z = stats.norm.ppf([(1 - .89) / 2, 1 - (1 - 0.89) / 2])
print("89% confidence interval:", _26.mean_p['p'] + _26.std_q * _26.z)
# # Medium
# ## 2M1
