#### David Ruhe

Ph.D. Student in Machine Learning

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# Normalizing Flows for Hierarchical Bayesian Analysis: A Gravitational Wave Population Study

##### Abstract

We propose parameterizing the population distribution of the gravitational wave population modeling framework (Hierarchical Bayesian Analysis) with a normalizing flow. We first demonstrate the merit of this method on illustrative experiments and then analyze four parameters of the latest LIGO data release: primary mass, secondary mass, redshift, and effective spin. Our results show that despite the small and notoriously noisy dataset, the posterior predictive distributions (assuming a prior over the parameters of the flow) of the observed gravitational wave population recover structure that agrees with robust previous phenomenological modeling results while being less susceptible to biases introduced by less-flexible distribution models. Therefore, the method forms a promising flexible, reliable replacement for population inference distributions, even when data is highly noisy.

When LIGO and VIRGO make gravitational wave detections (denoted ${x_i}$), they perform inference to obtain the associated physics parameters ${\theta_i^j}$. Typical parameters include the masses of the binary merger, spins, and redshift. Compared to the raw data ${x_i}$, these parameters are interpretable. However, there is much uncertainty about their values (Figure 2 on the left). After all: the event has happened millions of lightyears away, and our measurement devices are not precise enough to precisely measure the masses, spins, etc. Still, by royally sampling the posterior distribution $q(\theta|x_i)$, we get a good grasp of the interval in which these parameters probably live.