We present an accelerated pipeline, based on high-performance computing techniques and normalizing flows, for joint Bayesian parameter estimation and model selection and demonstrate its efficiency in gravitational wave astrophysics. We integrate the Jim inference toolkit, a normalizing flow-enhanced Markov chain Monte Carlo (MCMC) sampler, with the learned harmonic mean estimator. Our Bayesian evidence estimates run on $1$ GPU are consistent with traditional nested sampling techniques run on $16$ CPU cores, while reducing the computation time by factors of $5\times$ and $15\times$ for $4$-dimensional and $11$-dimensional gravitational wave inference problems, respectively. Our code is available in well-tested and thoroughly documented open-source packages, ensuring accessibility and reproducibility for the wider research community.

Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of the marginal likelihood. The learned harmonic mean estimator learns an importance sampling target distribution that approximates the optimal distribution. While the approximation need not be highly accurate, it is critical that the probability mass of the learned distribution is contained within the posterior in order to avoid the exploding variance problem. In previous work a bespoke optimization problem is introduced when training models in order to ensure this property is satisfied. In the current article we introduce the use of normalizing flows to represent the importance sampling target distribution. A flow-based model is trained on samples from the posterior by maximum likelihood estimation. Then, the probability density of the flow is concentrated by lowering the variance of the base distribution, i.e. by lowering its "temperature", ensuring its probability mass is contained within the posterior. This approach avoids the need for a bespoke optimisation problem and careful fine tuning of parameters, resulting in a more robust method. Moreover, the use of normalizing flows has the potential to scale to high dimensional settings. We present preliminary experiments demonstrating the effectiveness of the use of flows for the learned harmonic mean estimator. The harmonic code implementing the learned harmonic mean, which is publicly available, has been updated to now support normalizing flows.

Computing the marginal likelihood (ML) of a model requires marginalizing out all of the parameters and latent variables, a difficult high-dimensional summation or integration problem. To make matters worse, it is often hard to measure the accuracy of one's ML estimates. We present bidirectional Monte Carlo, a technique for obtaining accurate log-ML estimates on data simulated from a model. This method obtains stochastic lower bounds on the log-ML using annealed importance sampling or sequential Monte Carlo, and obtains stochastic upper bounds by running these same algorithms in reverse starting from an exact posterior sample. The true value can be sandwiched between these two stochastic bounds with high probability. Using the ground truth log-ML estimates obtained from our method, we quantitatively evaluate a wide variety of existing ML estimators on several latent variable models: clustering, a low rank approximation, and a binary attributes model. These experiments yield insights into how to accurately estimate marginal likelihoods.

Compressed Counting (CC) [22] was recently proposed for estimating the ath frequency moments of data streams, where 0 < a <= 2. CC can be used for estimating Shannon entropy, which can be approximated by certain functions of the ath frequency moments as a -> 1. Monitoring Shannon entropy for anomaly detection (e.g., DDoS attacks) in large networks is an important task. This paper presents a new algorithm for improving CC. The improvement is most substantial when a -> 1--. For example, when a = 0:99, the new algorithm reduces the estimation variance roughly by 100-fold. This new algorithm would make CC considerably more practical for estimating Shannon entropy. Furthermore, the new algorithm is statistically optimal when a = 0.5.

Many tasks (e.g., clustering) in machine learning only require the l

Many tasks (e.g., clustering) in machine learning only require the l

Many tasks (e.g., clustering) in machine learning only require the l