Carron & Szapudi (2014)

The big idea¶

Real surveys deliver points, not a continuous field, so the elegant continuous-field results of Carron & Szapudi (2013) need a discreteness-aware version. Carron & Szapudi solve the one-point case exactly and sketch the multipoint case for the Poisson sampling of an underlying lognormal density field — the same generative model as the FDF (lognormal-type field → local Poisson star counts). They show the corresponding optimal non-linear transformation is directly related to the maximum-a-posteriori Bayesian reconstruction of the underlying continuous field with a lognormal prior. The practical payoff is that “simple recipes for realizing the sufficient observables can be built on previously proposed algorithms” — the log/Gaussianizing transforms already implemented and tested in simulations.

Use in progenax¶

• Counts-in-cells sufficiency. This is the discrete analogue that licenses progenax’s CIC block: the stellar counts are a local-Poisson sampling of a lognormal-type field (the Szapudi & Pan (2004) CIC relation), and for such a field the sufficient observables are the ones the Differentiable inference — natal cloud parameters from cluster substructure likelihood already uses (the count distribution and its two-point clustering), not high-order count moments.

• Connects the count likelihood to a principled estimator rather than an ad hoc summary — the count_loglike block sits in this lineage.

Notes¶

• Together with Carron & Szapudi (2013) (continuous), Neyrinck, Szapudi & Szalay (2009) and Neyrinck, Szapudi & Szalay (2011) (empirical, matter and galaxies), this completes the argument that log-density 1pt + 2pt statistics are sufficient and information-optimal for a lognormal-type field — continuous or Poisson-sampled.

• The lognormal prior here is the pure-lognormal case; BM19 adds the self-gravity power-law tail, which progenax handles with the peaks-over-threshold tail block rather than a moment expansion.