Ultraplace

Code Structure

Goal

The goal of this project is two fold:

On the one hand, we want to study the how different models for photometric uncertainty propagates to our theory predictions.
On the other hand, we want to study if the method used to marginalise over said uncertainties matters; i.e. can we get away with analytical marginalisation.

Motivation

One of the leading contributions to the error budgets of cosmological analysis is the uncertainty in apparent redshifts of galaxies (also known as the radial galaxy distribution). This is particularly important when photometric catalogues, with course SED estimates, are used. Thus propagating said uncertainties to the final products of the analysis (cosmological constraints, ... etc) is paramount.

The uncertainties in the radial distribution of galaxies is naturally expressed in the form of a statistical process. However, due the finite number of galaxies at a given position, the process is often discretised as a histogram with finite bins and an associated covariance matrix. The number of discrete bins is normally of order 10^2. This large number of parameters makes propagating their impact computationally prohibitive for traditional inference methods. Thus, cosmological analysis have so attempted to summarise the radial distribution in terms of far lower number of parameters (10^1).

Literature review

Papers

Analytic marginalization over CMB calibration and beam uncertainty by Bridle et al (0112114) - 2001
Analytic Methods for Cosmological Likelihoods by Taylor & Kitchin (1003.1136) - 2011
Self-calibration and robust propagation of photometric redshift distribution uncertainties in weak gravitational lensing by Stölzner et al (2012.07707) - 2021
Analytical marginalisation over photometric redshift uncertainties in cosmic shear analyses by Ruiz-Zapatero et al (2301.11978) - 2023

Photo-z Models

Shifts: $$p(z + \Delta z)$$
Shifts & widths: $$p(z_c + w_{z}(z-z_c) + \Delta z)$$
Eigen-functions: $$p(z) = \sum_i^n \lambda_i \phi_i(z)$$
Comb: $$p(z) = \sum_i^{N_z} A_i , \mathcal{N}(z; z_i, \sigma^2)$$
Full Histogram: $$p(z) = \boldsymbol{n}$$
Neuronal Network: $$p(z) = NN(\boldsymbol{\alpha})$$

Stölzner vs Ruiz-Zapatero

	Stölzner	Ruiz-Zapatero
Photo-z model	Comb	No model (full histogram)
Marginalisation	Laplace approximation	Laplace approximation (optimised for linear parameters)
Assumptions	Nuisance posteriors are somewhat Gaussian (base Laplace assumption)	Nuisance posteriors are somewhat Gaussian (base Laplace assumption)
		Gaussian Likelihood
		Photo-z's are tightly constraint such that their impact of the theory can be linearized
		Gaussian priors for the Photo-z's
Final expression	$P(\Omega \| d) = P(d \| \Omega, n^\star) + \ln(Det(F_{n^\star}))$	$P(\Omega \| d) = (d - t^\star)^T \tilde{C}^{-1} (d-t^\star) - 2 \ln P(\Omega) + \ln(Det(T^T C^{-1}T+C_n^{-1}))$ where $\tilde{C} = C + TC_nT^T$ and $T= \partial t/\partial n$
Advantages	More general	Best-fit nuisance parameters can be found analytically
		Laplace term simplifies to $\ln(Det(C_n^{-1}))$
		Profile term becomes an extra contribution to the data covariance.
		Faster

JaimeRZP / ultraplace