tinyda 0.9.5

Delayed Acceptance MCMC Sampler

tinyDA

Delayed Acceptance (Christen & Fox, 2005) 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.

It is fully imperative and easy to use!

Installation

tinyDA can be installed from PyPI:

pip install tinyDA

Features

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)
• Adaptive pCN - Hu and Yao (2016)
• 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

• A bunch of plotting functions
• Rank-normalised split- and ESS - Vehtari et al. (2020)

Dependencies:

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

Usage

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.LogLike(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.LinkFactory

At the heart of the TinyDA sampler sits what we call a LinkFactory, 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 model output, given the parameters.
3. Constructing tda.Link instances that hold information for each sample.

The LinkFactory must be defined by inheritance from either tda.LinkFactory or tda.BlackBoxLinkFactory. The former allows for computing the model output directly from the input parameters, using pure Python or whichever external library you want to call. The evaluate_model() method must thus be overwritten:

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

my_link_factory = MyLinkFactory(my_prior, my_loglike)

The latter allows for feeding some model object to the LinkFactory at initialisation, which is then assigned as a class attribute. This is useful for e.g. PDE solvers. Your model must return ordered data when called (e.g. via a __call__(self, parameters) method), and there is no need to overwrite evaluate_model(). This is what happens under the hood:

class BlackBoxLinkFactory(LinkFactory):
    def evaluate_model(self, parameters):
            
        # get the model output.
        model_output = self.model(parameters)
        
        # get the quantity of interest.
        if self.get_qoi:
            qoi = self.model.get_qoi()
        else:
            qoi = None
            
        # return everything.
        return model_output, qoi

my_link_factory = BlackBoxLinkFactory(my_model, my_prior, my_loglike, get_qoi=True)

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 link factory, my_link_factory_coarse, and a fine link factory my_link_factory_fine, the Delayed Acceptance sampler can be run using tinyDA.sample():

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

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.

TODO

• Parallel multi-chain sampling
• More user-friendly diagnostics
• Multilevel Delayed Acceptance