Debiasing inference in large-scale structure with non-flat volume measures

Increasingly large parameter spaces, used to more accurately model precision observables in physics, can paradoxically lead to large deviations in the inferred parameters of interest -- a bias known as volume projection effects -- when marginalising over many nuisance parameters. For posterior distributions that admit a Laplace expansion, we show that this artefact of Bayesian inference can be mitigated by defining expectation values with respect to a non-flat volume measure, such that the posterior mean becomes unbiased on average. We begin by finding a measure that ensures the mean is an unbiased estimator of the mode. Although the mode itself, as we rediscover, is biased under sample averaging, this choice yields the least biased estimator due to a cancellation we clarify. We further explain why bias in marginal posteriors can appear relatively large, yet remains correctable, when the number of nuisances is large. To demonstrate our approach, we present mock analyses in large-scale structure (LSS) wherein cosmological parameters are subject to large projection effects (at the 1-2$\sigma$ level) under a flat measure, that are however recovered at high fidelity ($<0.1\sigma$) when estimated using non-flat counterparts. Our cosmological analyses are enabled by $\texttt{PyBird-JAX}$, a fast, differentiable pipeline for LSS developed in our companion paper [1].