ULA#

class cuqi.experimental.mcmc.ULA(target=None, scale=1.0, **kwargs)#

Unadjusted Langevin algorithm (ULA) (Roberts and Tweedie, 1996)

It approximately samples a distribution given its logpdf gradient based on the Langevin diffusion dL_t = dW_t + 1/2*Nabla target.logd(L_t)dt, where W_t is the dim-dimensional standard Brownian motion. ULA results from the Euler-Maruyama discretization of this Langevin stochastic differential equation (SDE).

For more details see: Roberts, G. O., & Tweedie, R. L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, 341-363.

Parameters:

target (cuqi.distribution.Distribution) – The target distribution to sample. Must have logd and gradient method. Custom logpdfs and gradients are supported by using a cuqi.distribution.UserDefinedDistribution.
initial_point (ndarray) – Initial parameters. Optional
scale (float) – The Langevin diffusion discretization time step (In practice, scale must be smaller than 1/L, where L is the Lipschitz of the gradient of the log target density, logd).
callback (callable, Optional) – If set this function will be called after every sample. The signature of the callback function is callback(sample, sample_index), where sample is the current sample and sample_index is the index of the sample. An example is shown in demos/demo31_callback.py.

Example

# Parameters
dim = 5 # Dimension of distribution
mu = np.arange(dim) # Mean of Gaussian
std = 1 # standard deviation of Gaussian

# Logpdf function
logpdf_func = lambda x: -1/(std**2)*np.sum((x-mu)**2)
gradient_func = lambda x: -2/(std**2)*(x - mu)

# Define distribution from logpdf and gradient as UserDefinedDistribution
target = cuqi.distribution.UserDefinedDistribution(dim=dim, logpdf_func=logpdf_func,
    gradient_func=gradient_func)

# Set up sampler
sampler = cuqi.experimental.mcmc.ULA(target, scale=1/dim**2)

# Sample
sampler.sample(2000)

A Deblur example can be found in demos/demo27_ULA.py # TODO: update demo once sampler merged

__init__(target=None, scale=1.0, **kwargs)#

Initializer for abstract base class for all samplers.

Any subclassing samplers should simply store input parameters as part of the __init__ method.

The actual initialization of the sampler should be done in the _initialize method.

Parameters:

target (cuqi.density.Density) – The target density.
initial_point (array-like, optional) – The initial point for the sampler. If not given, the sampler will choose an initial point.
callback (callable, optional) – A function that will be called after each sample is drawn. The function should take two arguments: the sample and the index of the sample. The sample is a 1D numpy array and the index is an integer. The callback function is useful for monitoring the sampler during sampling.

Methods

`__init__`([target, scale])	Initializer for abstract base class for all samplers.
`get_history`()	Return the history of the sampler.
`get_samples`()	Return the samples.
`get_state`()	Return the state of the sampler.
`initialize`()	Initialize the sampler by setting and allocating the state and history before sampling starts.
`load_checkpoint`(path)	Load the state of the sampler from a file.
`reinitialize`()	Re-initialize the sampler.
`sample`(Ns[, batch_size, sample_path])	Sample Ns samples from the target density.
`save_checkpoint`(path)	Save the state of the sampler to a file.
`set_history`(history)	Set the history of the sampler.
`set_state`(state)	Set the state of the sampler.
`step`()	Perform one step of the sampler by transitioning the current point to a new point according to the sampler's transition kernel.
`tune`(skip_len, update_count)	Tune the parameters of the sampler.
`validate_target`()	Validate the target is compatible with the sampler.
`warmup`(Nb[, tune_freq])	Warmup the sampler by drawing Nb samples.

Attributes

`dim`	Dimension of the target density.
`geometry`	Geometry of the target density.
`target`	Return the target density.