Posterior Distributions

class oktopus.posterior.Posterior(likelihood, prior)

Defines a posterior distribution.

Examples

>>> import math
>>> from oktopus import PoissonPosterior, PoissonLikelihood
>>> import autograd.numpy as np
>>> np.random.seed(0)
>>> toy_data = np.random.randint(1, 20, size=100)
>>> def mean(l):
...     return np.array([l])
>>> logL = PoissonLikelihood(data=toy_data, mean=mean)
>>> logP = Posterior(likelihood=logL, prior=logL.jeffreys_prior)
>>> mean_hat = logP.fit(x0=10.5)
>>> mean_hat.x
array([ 9.28500001])
>>> print(np.mean(toy_data)) # MLE estimate
9.29

Attributes

likelihood	(callable or instance of :class:oktopus.Likelihood) If callable, must provide a method called evaluate which returns the negative of the log likelihood.
prior	(callable or instance of :class:oktopus.Prior) If callable, must provide a method called evaluate which returns the negative of the log of the distribution.

Methods

__call__(params)	Calls evaluate()
evaluate(params)	Evaluates the negative of the log of the posterior at params.
fit([optimizer])	Minimizes the evaluate() function using scipy.optimize.minimize(), scipy.optimize.differential_evolution(), scipy.optimize.basinhopping(), or skopt.gp.gp_minimize().
gradient(params)	Evaluates the gradient of the negative of the log of the posterior at params.
hessian(params)	Returns the Hessian matrix of the loss function evaluated at params

evaluate(params)

Evaluates the negative of the log of the posterior at params.

Parameters:

params : scalar or array-like

Value at which the posterior will be evaluated

Returns:

value : scalar

Value of the negative of the log of the posterior at params

gradient(params)

Evaluates the gradient of the negative of the log of the posterior at params.

Parameters:

params : scalar or array-like

Value at which the posterior will be evaluated

Returns:

value : scalar

Value of the negative of the log of the posterior at params

class oktopus.posterior.GaussianPosterior(data, mean, var, prior)

Implements the negative log posterior distribution for uncorrelated (possibly non identically) distributed Gaussian measurements with known variances.

Examples

>>> from oktopus import GaussianPosterior, GaussianPrior, UniformPrior, JointPrior
>>> import autograd.numpy as np
>>> #from matplotlib import pyplot as plt
>>> x = np.linspace(0, 10, 200)
>>> np.random.seed(0)
>>> fake_data = x * 3 + 10 + np.random.normal(scale=2, size=x.shape)
>>> def line(x, slope, intercept):
...     return slope * x + intercept
>>> my_line = lambda slope, intercept: line(x, slope, intercept)
>>> slope_prior = UniformPrior(lb=1., ub=10.)
>>> intercept_prior = UniformPrior(lb=5., ub=20.)
>>> joint_prior = JointPrior(slope_prior, intercept_prior)
>>> logP = GaussianPosterior(data=fake_data, mean=my_line, var=4, prior=joint_prior)
>>> p0 = (slope_prior.mean, intercept_prior.mean) # initial guesses for slope and intercept
>>> p_hat = logP.fit(x0=p0, method='powell')
>>> p_hat.x # fitted parameters
array([  2.96264088,  10.3286166 ])
>>> #plt.plot(x, fake_data, 'o')
>>> #plt.plot(x, line(*p_hat.x))
>>> # The exact values from linear algebra are:
>>> M = np.array([[np.sum(x * x), np.sum(x)], [np.sum(x), len(x)]])
>>> slope, intercept = np.dot(np.linalg.inv(M), np.array([np.sum(fake_data * x), np.sum(fake_data)]))
>>> print(slope)
2.96264087528
>>> print(intercept)
10.3286166099

Attributes

data	(ndarray) Observed data
mean	(callable) Mean model
var	(scalar or array-like) Uncertainties on the observed data.
prior	(callable) Negative log prior as a function of the parameters See UniformPrior

Methods

__call__(params)	Calls evaluate()
evaluate(params)	Evaluates the negative of the log of the posterior at params.
fit([optimizer])	Minimizes the evaluate() function using scipy.optimize.minimize(), scipy.optimize.differential_evolution(), scipy.optimize.basinhopping(), or skopt.gp.gp_minimize().
gradient(params)	Evaluates the gradient of the negative of the log of the posterior at params.
hessian(params)	Returns the Hessian matrix of the loss function evaluated at params

class oktopus.posterior.PoissonPosterior(data, mean, prior)

Implements the negative of the log posterior distribution for independent (possibly non-identically) distributed Poisson measurements.

Examples

>>> import math
>>> from oktopus import PoissonPosterior, UniformPrior, GaussianPrior
>>> import autograd.numpy as np
>>> np.random.seed(0)
>>> toy_data = np.random.randint(1, 20, size=100)
>>> def mean(l):
...     return np.array([l])
>>> logP = PoissonPosterior(data=toy_data, mean=mean, prior=UniformPrior(lb=1., ub=20.))
>>> mean_hat = logP.fit(x0=10.5)
>>> mean_hat.x # MAP is the same of MLE for uniform prior
array([ 9.29000013])
>>> logP = PoissonPosterior(data=toy_data, mean=mean, prior=GaussianPrior(mean=10, var=4))
>>> mean_hat = logP.fit(x0=10.5)
>>> mean_hat.x
array([ 9.30614261])

Attributes

data	(ndarray) Observed count data
mean	(callable) Mean of the Poisson distribution Note: If you would like to get uncertainties by using the uncertainties method, then this model must be defined with autograd numpy wrapper
prior	(callable) Negative log prior as a function of the parameters. See UniformPrior

Methods

__call__(params)	Calls evaluate()
evaluate(params)	Evaluates the negative of the log of the posterior at params.
fit([optimizer])	Minimizes the evaluate() function using scipy.optimize.minimize(), scipy.optimize.differential_evolution(), scipy.optimize.basinhopping(), or skopt.gp.gp_minimize().
gradient(params)	Evaluates the gradient of the negative of the log of the posterior at params.
hessian(params)	Returns the Hessian matrix of the loss function evaluated at params

class oktopus.posterior.MultivariateGaussianPosterior(data, mean, cov, dim, prior)

Implements the posterior distribution for a multivariate gaussian distribution.

Attributes

data	(ndarray) Observed data
mean	(callable) Mean model
cov	(ndarray or callable) If callable, the parameters of the covariance matrix will be fitted.
dim	(int) Number of parameters of the mean model.
prior	(callable) Negative log prior as a function of the parameters. See :class:UniformPrior

Methods

__call__(params)	Calls evaluate()
evaluate(params)	Evaluates the negative of the log of the posterior at params.
fit([optimizer])	Minimizes the evaluate() function using scipy.optimize.minimize(), scipy.optimize.differential_evolution(), scipy.optimize.basinhopping(), or skopt.gp.gp_minimize().
gradient(params)	Evaluates the gradient of the negative of the log of the posterior at params.
hessian(params)	Returns the Hessian matrix of the loss function evaluated at params

Inheritance Diagram