tinyda 0.9.15

Delayed Acceptance MCMC Sampler

tinyDA

Multilevel Delayed Acceptance MCMC sampler with finite-length subchain sampling and adaptive error modelling. This is intended as a simple, lightweight implementation, with minimal dependencies, i.e. nothing beyond the SciPy stack and ArviZ. It is fully imperative and easy to use!

For instructions, have a look at the documentation, the examples or the usage section below.

Installation

tinyDA can be installed from PyPI:

pip install tinyda

Dependencies

• NumPy
• SciPy
• ArviZ
• tqdm
• Ray (multiprocessing, optional)

Features

Samplers

• Metropolis-Hastings
• Delayed Acceptance (Christen & Fox, 2005)
• Multilevel Delayed Acceptance (Lykkegaard et al. 2022)

Proposals

• Random Walk Metropolis Hastings (RWMH) - Metropolis et al. (1953), Hastings (1970)
• preconditioned Crank-Nicolson (pCN) - Cotter et al. (2013)
• Adaptive Metropolis (AM) - Haario et al. (2001)
• Operator-weighted pCN - Law (2014)
• Metropolis Adjusted Langevin Algorithm (MALA) - Roberts & Tweedie (1996)
• DREAM(Z) - Vrugt (2016)
• Multiple-Try Metropolis (MTM) - Liu et al. (2000)

Adaptive Error Models

• State independent - Cui et al. (2018)
• State dependent - Cui et al. (2018)

Diagnostics

• Convert a tinyDA chain to an ArviZ InferenceData object for near-unlimited diagnostics!

Usage

Documentation is available at Read the Docs. A few illustrative examples are available as Jupyter Notebooks in the root directory. Below is a short summary of the core features.

Distributions

The prior and likelihood can be defined using standard scipy.stats classes:

import tinyDA as tda

from scipy.stats import multivariate_normal

# set the prior mean and covariance.
mean_prior = np.zeros(n_dim)
cov_prior = np.eye(n_dim)

# set the covariance of the likelihood.
cov_likelihood = sigma**2*np.eye(data.shape[0])

# initialise the prior distribution and likelihood.
my_prior = multivariate_normal(mean_prior, cov_prior)
my_loglike = tda.GaussianLogLike(data, cov_likelihood)

If using a Gaussian likelihood, we recommend using the tinyDA implementation, since it is unnormalised and plays along well with tda.AdaptiveLogLike used for the Adaptive Error Model. Home-brew distributions can easily be defined, and must have a .rvs() method for drawing random samples and a logpdf(x) method for computing the log-likelihood, as per the SciPy implementation.

tinyDA.Posterior

The heart of the TinyDA sampler is the tinyDA.Posterior, which is responsible for:

1. Calling the model with some parameters (a proposal) and collecting the model output.
2. Evaluating the prior density of the parameters, and the likelihood of the data, given the parameters.
3. Constructing tda.Link instances that hold information for each sample.

The tinyDA.Posterior takes as input the prior, the likelihood, and a forward model. Therefore, a forward model must be defined. This model can be either a function model_output = my_function(parameters) or a class instance with a .__call__(self, parameters) method. The function or __call__ method must return either just the model output or a tuple of (model_output, qoi). In this example, we define a class that performs simple linear regression on whatever inputs x we have.

class MyLinearModel:
    def __init__(self, x):

        self.x = x
        
    def __call__(self, parameters):
        
        # the model output is a simple linear regression
        model_output = parameters[0] + parameters[1]*self.x
        
        # no quantity of interest beyond the parameters.
        qoi = None
        
        # return both.
        return model_output, qoi

my_model = MyLinearModel(x)
my_posterior = tda.Posterior(my_prior, my_loglike, my_model)

Proposals

A proposal is simply initialised with its parameters:

# set the covariance of the proposal distribution.
am_cov = np.eye(n_dim)

# set the number of iterations before starting adaptation.
am_t0 = 1000

# set some adaptive metropolis tuning parameters.
am_sd = 1
am_epsilon = 1e-6

# initialise the proposal.
my_proposal = tda.AdaptiveMetropolis(C0=am_cov, t0=am_t0, sd=am_sd, epsilon=am_epsilon)

Sampling

After defining a proposal, a coarse posterior my_posterior_coarse, and a fine posterior my_posterior_fine, the Delayed Acceptance sampler can be run using tinyDA.sample():

my_chains = tda.sample([my_posterior_coarse, my_posterior_fine], 
                       my_proposal, 
                       iterations=12000, 
                       n_chains=2, 
                       subsampling_rate=10)

If using a hirarchy with more than two models, a Multilevel Delayed Acceptance sampler can be run by supplying a list of posteriors in ascending order and a correponsing list of subsampling rates:

my_chains = tda.sample([my_posterior_level0, 
                        my_posterior_level1, 
                        my_posterior_level2, 
                        my_posterior_level3], 
                       my_proposal, 
                       iterations=12000, 
                       n_chains=2, 
                       subsampling_rate=[10, 5, 5])

Postprocessing

The entire sampling history is now stored in my_chains in the form of a dictionary with tinyDA.Link instances. You can convert the output of tinyDA.sample() to an ArviZ InferenceData object with

idata = tda.to_inference_data(my_chains, burnin=2000)

If you want to have a look at the coarse samples, you can pass an additional argument:

idata = tda.to_inference_data(my_chains, level='coarse', burnin=20000)

The idata object can then be used with the ArviZ diagnostics suite to e.g. get MCMC statistics, plot the traces and so on.

Contributing

If you feel that tinyDA is missing some features, or that something could be improved, please do not hesitate to create a fork and submit a PR! If you want to help improve the package, please have a look at the issues and consider if something seems doable to you.

If you would like to contribute, please consider the following:

• It's called tinyDA because it's small. The list of dependencies should be kept short. Great things can be achieved using NumPy!
• tinyDA has loads of nice features, but it's somewhat lacking in terms of CI. Any kind of CI, tests and improvements to the software infrastructure would be greatly appreciated!

The development of tinyDA is sponsored by digiLab.

TODO

• Parallel multi-chain sampling
• More user-friendly diagnostics
• Multilevel Delayed Acceptance
• MALA proposal
• Tests
• Variance Reduction
• Wrapper for framework-agnostic adaptive coarse model
• Embedded spaces for hierachical models