# Predicting conditional probability distributions of redshifts of Active Galactic Nuclei using Hierarchical Correlation Reconstruction

arXiv: https://arxiv.org/abs/2206.06194 · HTML: https://mlsanity.com/paper/2206.06194/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 9 references that are in the map.
- First posted on arXiv 13 Jun 2022.

## Builds on (most cited first, 2 of 2)

- [Hierarchical correlation reconstruction with missing data, for example for biology-inspired neuron](https://mlsanity.com/paper/1804.06218/) (2018)
- [Modelling bid-ask spread conditional distributions using hierarchical correlation reconstruction](https://mlsanity.com/paper/1911.02361/) (2019)

## Built on by (most cited first, 4 of 4)

- [Biology-inspired joint distribution neurons based on Hierarchical Correlation Reconstruction allowing for multidirectional propagation of values and densities](https://mlsanity.com/paper/2405.05097/) (2024)
- [Low cost prediction of probability distributions of molecular properties for early virtual screening](https://mlsanity.com/paper/2207.11174/) (2022)
- [Extracting individual variable information for their decoupling, direct mutual information and multi-feature Granger causality](https://mlsanity.com/paper/2311.13431/) (2023)
- [Linear cost mutual information estimation and independence test of similar performance as HSIC](https://mlsanity.com/paper/2508.18338/) (2025)

## Abstract (arXiv)

While there is a general focus on prediction of values, real data often only allows to predict conditional probability distributions, with capabilities bounded by conditional entropy H(Y|X). If additionally estimating uncertainty, we can treat a predicted value as the center of Gaussian of Laplace distribution - idealization which can be far from complex conditional distributions of real data. This article applies Hierarchical Correlation Reconstruction (HCR) approach to inexpensively predict quite complex conditional probability distributions (e.g. multimodal): by independent MSE estimation of multiple moment-like parameters, which allow to reconstruct the conditional distribution. Using linear regression for this purpose, we get interpretable models: with coefficients describing contributions of features to conditional moments. This article extends on the original approach especially by using Canonical Correlation Analysis (CCA) for feature optimization and l1 "lasso" regularization, focusing on practical problem of prediction of redshift of Active Galactic Nuclei (AGN) based on Fourth Fermi-LAT Data Release 2 (4LAC) dataset.