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The sampler interface

Note: this tutorial requires that the user is already familiar with the flexible interface.

sbi implements three methods: SNPE, SNLE, and SNRE. When using SNPE, the trained neural network directly approximates the posterior. Thus, sampling from the posterior can be done by sampling from the trained neural network. The neural networks trained in SNLE and SNRE approximate the likelihood(-ratio). Thus, in order to draw samples from the posterior, one has to perform additional sampling steps, e.g. Markov-chain Monte-Carlo (MCMC). In sbi, the implemented samplers are:

  • Markov-chain Monte-Carlo (MCMC)

  • Rejection sampling

  • Variational inference (VI)

When using the flexible interface, the sampler as well as its attributes can be set with sample_with="mcmc", mcmc_method="slice_np", and mcmc_parameters={}. However, for full flexibility in customizing the sampler, we recommend using the sampler interface. This interface is described here. Further details can be found here.

Main syntax for SNLE

import torch

from sbi.inference import SNLE
from sbi.inference import likelihood_estimator_based_potential, MCMCPosterior

# dummy Gaussian simulator for demonstration
num_dim = 2
prior = torch.distributions.MultivariateNormal(torch.zeros(num_dim), torch.eye(num_dim))
theta = prior.sample((1000,))
x = theta + torch.randn((1000, num_dim))
x_o = torch.randn((1, num_dim))

inference = SNLE(show_progress_bars=False)
likelihood_estimator = inference.append_simulations(theta, x).train()

potential_fn, parameter_transform = likelihood_estimator_based_potential(
    likelihood_estimator, prior, x_o
)
posterior = MCMCPosterior(
    potential_fn, proposal=prior, theta_transform=parameter_transform
)

 Neural network successfully converged after 52 epochs.

Further explanation

The first lines are the same as for the flexible interface:

inference = SNLE()
likelihood_estimator = inference.append_simulations(theta, x).train()

 Neural network successfully converged after 33 epochs.

Next, we obtain the potential function. A potential function is a function of the parameter \(f(\theta)\). The posterior is proportional to the product of likelihood and prior: \(p(\theta | x_o) \propto p(x_o | \theta)p(\theta)\). The potential function is the logarithm of the right-hand side of this equation: \(f(\theta) = \log(p(x_o | \theta)p(\theta))\)

potential_fn, parameter_transform = likelihood_estimator_based_potential(
    likelihood_estimator, prior, x_o
)

By calling the potential_fn, you can evaluate the potential:

# Assuming that your parameters are 1D.
potential = potential_fn(
    torch.zeros(1, num_dim)
)  # -> returns f(0) = log( p(x_o|0) p(0) )

The other object that is returned by likelihood_estimator_based_potential is a parameter_transform. The parameter_transform is a pytorch transform. The parameter_transform is a fixed transform that is can be applied to parameter theta. It transforms the parameters into unconstrained space (if the prior is bounded, e.g. BoxUniform), and standardizes the parameters (i.e. zero mean, one std). Using parameter_transform during sampling is optional, but it usually improves the performance of MCMC.

theta_tf = parameter_transform(torch.zeros(1, num_dim))
theta_original = parameter_transform.inv(theta_tf)
print(theta_original)  # -> tensor([[0.0]])

tensor([[0., 0.]])

After having obtained the potential_fn, we can sample from the posterior with MCMC or rejection sampling:

from sbi.inference import MCMCPosterior, RejectionPosterior

posterior = MCMCPosterior(
    potential_fn, proposal=prior, theta_transform=parameter_transform
)
posterior = RejectionPosterior(potential_fn, proposal=prior)

Main syntax for SNPE

SNPE usually does not require MCMC or rejection sampling (if you still need it, you can use the same syntax as above with the posterior_estimator_based_potential function). Instead, SNPE samples from the neural network. If the support of the prior is bounded, some samples can lie outside of the support of the prior. The DirectPosterior class automatically rejects these samples:

from sbi.inference import SNPE
from sbi.inference import DirectPosterior

inference = SNPE()
posterior_estimator = inference.append_simulations(theta, x).train()

posterior = DirectPosterior(posterior_estimator, prior=prior)

 Neural network successfully converged after 57 epochs.