Log Probability Differences and Posterior Discrepancies in SNLE/SNPE with Various Sampling Techniques

[phelipedarc]

Hi,
I’m currently training an SNLE and a (T)SNPE aiming to obtain samples from the posterior and their corresponding log probabilities. My main goal is to follow the approach outlined by Spurio Mancini in Bayesian Model Comparison for Simulation-Based Inference (2023) by using a Learnt Harmonic Mean estimator to compute the evidence.

I've noticed that it's possible to calculate the log probability from the trained NPE (using samples generated by the NPE) and from the trained NLE (using samples from either MCMC, Variational Inference (VI), or the NPE-generated samples).
Example Code:

# Log prob from the NPE
posterior_samples = posterior.sample((2500,), x=x_o)
posterior_samples = np.array(posterior_samples)
logprobsnpe = np.array(posterior.potential(posterior_samples, x=x_o))

# Log prob from the SNLE with SNPE samples
potential_fn, parameter_transform = likelihood_estimator_based_potential(density_estimator_snle, prior, x_o)
posteriormcmc = MCMCPosterior(potential_fn, proposal=prior, theta_transform=parameter_transform, warmup_steps=10, num_chains=50)
logprobmcmc = posteriormcmc.potential(posterior_samples, x=x_o)

# Log prob from the SNLE with SNLE samples using MCMC
posterior_samples_mcmc = posteriormcmc.sample((25000,), x=x_o)
logprobmcmc0 = posteriormcmc.potential(posterior_samples_mcmc, x=x_o)

My two questions are:

1. Log Probability Differences: What is the fundamental difference between the log probabilities calculated from the SNPE and SNLE for the same set of samples? Is there any specific reason for potential discrepancies, particularly, does the NPE and NLE have to converge to the ""same"" posterior distribution in order for my logprob values, using SNPE samples and the NLE density estimator, to be considered reliable?

2. Sampler Variations: When I use different sampling techniques (Rejection, MCMC, VI), I observe varying posterior distributions, with VI producing a notably narrower posterior. Could you explain to me why this discrepancy occurs, especially for VI, and how it might impact the posterior distribution or the evidence calculation when using Bayesian model comparison methods?

Any insights or explanations regarding these observations would be greatly appreciated!

Thanks in advance for your help!

Posterior using VI sampling

Posterior using Rejection sampling

[michaeldeistler]

Hi there! Thanks for opening this!

1. NPE returns log-probabilities of the normalized posterior p(t | x). NLE returns log-probabilities that are not normalized, i.e. it returns p(x|t)p(t) (without dividing by p(x)). So, overall, the log-probablities of NPE and NLE should be exactly proportional for a given observation, but are not expected to match exactly.

2. Generally, when you can use rejection sampling (because you have few parameters, typically less than 5), then it is probably your best choice. We have found that MCMC is also very robust and gives good results. Variational inference is a very fast method if you have (very) many parameters, but it can be inaccurate. In particular, VI (by default) is prone to giving too narrow posteriors. You can potentially overcome these issues with using a different objective, e.g. VIPosterior(..., vi_method="fKL"). However, again: If you can use rejection sampling or MCMC (and they are not prohibitively slow), then I would recommend that.

Hope this helps!
Michael