CosmoForge.PICSLike Package

PICSLike (Pixel-based Inference with Correlated-Skies Likelihood) is a pixel-based likelihood analysis package for spin-0 and spin-2 fields on the sphere. It provides tools for computing the likelihood of observations given theoretical predictions directly in pixel space, offering an alternative to harmonic-space methods. Applications include CMB temperature and polarization, galaxy surveys, and any other signal described by angular power spectra.

Overview

PICSLike implements pixel-based likelihood analysis, which is particularly useful for:

Incomplete sky coverage: Natural handling of masked regions without harmonic-space complications
Non-Gaussian features: Direct treatment of non-Gaussian signals and systematics
Cross-correlation analysis: Efficient computation of cross-correlations between different maps
Computational efficiency: Potentially faster for certain analysis configurations

Mathematical Foundation

Pixel-Based Likelihood

The pixel-based likelihood function is computed as:

\[\ln \mathcal{L}(\theta) = -\frac{1}{2} (\mathbf{d} - \mathbf{s}(\theta))^T \mathbf{C}^{-1} (\mathbf{d} - \mathbf{s}(\theta))\]

where:

\(\mathbf{d}\) is the observed data vector in pixel space
\(\mathbf{s}(\theta)\) is the theoretical signal prediction for parameters \(\theta\)
\(\mathbf{C} = \mathbf{S}(\theta) + \mathbf{N}\) is the total covariance matrix

The chi-squared statistic is:

\[\chi^2(\theta) = (\mathbf{d} - \mathbf{s}(\theta))^T \mathbf{C}^{-1} (\mathbf{d} - \mathbf{s}(\theta))\]

Key Features

Parameter Grid Evaluation

Systematic exploration of parameter space on a grid
Automatic signal covariance computation for each parameter point
MPI parallelization for efficient grid traversal
Support for arbitrary number of cosmological parameters

Statistical Analysis

Chi-squared and log-likelihood computation at each grid point
Best-fit parameter extraction from likelihood surface
Confidence interval estimation via likelihood marginalization
Support for multiple simulation realizations

High Performance Computing

MPI parallelization across parameter grid points
Memory-optimized covariance matrix operations
Integration with CosmoForge infrastructure

Package Contents

Quick Reference

The package provides comprehensive documentation for each component with mathematical foundations, computational details, and practical usage examples.

Usage Examples

Basic Pixel-Based Likelihood Analysis

from picslike import PICSLike

# Initialize with configuration file
picslike = PICSLike(params_file="config/pixel_analysis.yaml")

# Run complete analysis pipeline
picslike.run()

# Get results (master process only)
if picslike.rank == 0:
    chi2 = picslike.get_chi_squared()
    best_fit = picslike.get_best_fit()
    print(f"Best-fit parameters: {best_fit}")

Step-by-Step Pipeline

from picslike import PICSLike

# Initialize
picslike = PICSLike(params_file="config/analysis.yaml")

# Setup pipeline components
picslike.setup_parameter_grid()
picslike.setup_fields()
picslike.setup_geometry()
picslike.setup_covariance_matrices()
picslike.setup_cls()
picslike.setup_beams()
picslike.setup_maps()

# Compute likelihood across parameter grid
picslike.compute()

# Extract and save results
if picslike.rank == 0:
    picslike.save_results("output/likelihood_results.npz")

MPI Parallel Execution

# Run with 8 MPI processes
mpirun -n 8 python pixel_analysis.py config.yaml

Result Analysis

from picslike import LikelihoodResult

# Load saved results
result = LikelihoodResult.load("output/likelihood_results.npz")

# Get confidence intervals
intervals_68 = result.get_confidence_intervals(0.68)
intervals_95 = result.get_confidence_intervals(0.95)

# Marginalize over parameters
marg_omega_b = result.get_marginalized_likelihood('omega_b')

# Summary statistics
summary = result.get_summary_statistics()

Performance Considerations

Memory Requirements

The primary memory bottleneck is covariance matrix storage, scaling as \(O(N_{pix}^2)\). For typical analyses:

nside=256: ~4 GB for temperature + polarization
nside=512: ~16 GB for temperature + polarization
nside=1024: ~250 GB for temperature + polarization

Computational Scaling

Likelihood evaluation: \(O(N_{param} \times N_{pix}^3)\) per parameter point
Matrix inversion: Dominant computational cost
MPI parallelization: Parameter grid points distributed across processes

Optimization Strategies

Use appropriate HEALPix resolution (nside) for science goals
Pre-compute theoretical spectra grids for efficiency
Consider reduced covariance matrices for large-scale analyses
Leverage MPI parallelization for compute-bound operations

Configuration

PICSLike uses YAML configuration files for analysis setup:

# HEALPix parameters
nside: 512
lmax: 1000

# Field configuration
spins: [0, 2]
labels: ["T", "E", "B"]
physical_labels: ["T", "Q", "U"]

# Simulation settings
nsims: 100

# Parameter grid
parameters:
  omega_b:
    min: 0.020
    max: 0.025
    n_points: 11
  omega_c:
    min: 0.10
    max: 0.14
    n_points: 11

# Input files
mapsfile1: "data/observed_maps.fits"
covmatfile1: "data/noise_covariance.bin"
clfile: "theory/theoretical_spectra.txt"
beamfile: "data/beam_transfer.fits"

Integration with CosmoForge

PICSLike seamlessly integrates with other CosmoForge packages:

cosmocore: Provides base functionality, field management, and mathematical utilities
qube: Complementary QML power spectrum analysis for comparison
meta: Workflow management and configuration utilities

References

[Wandelt2004]

Wandelt, B.D., Larson, D.L. & Lakshminarayanan, A. “Global, exact cosmic microwave background data analysis using Gibbs sampling” Phys. Rev. D 70, 083511 (2004)

[Jewell2004]

Jewell, J., Levin, S. & Anderson, C.H. “Application of Monte Carlo algorithms to the Bayesian analysis of the cosmic microwave background” Astrophys. J. 609, 1-14 (2004)

[Eriksen2004]

Eriksen, H.K. et al. “Power Spectrum Estimation from High-Resolution Maps by Gibbs Sampling” Astrophys. J. Suppl. 155, 227-241 (2004)

[Planck2020]

Planck Collaboration “Planck 2018 results. V. CMB power spectra and likelihoods” Astron. Astrophys. 641, A5 (2020)