Learn Markov Chain Monte Carlo with Me.

In this page, I provide several practical .ipynb notebooks designed to help researchers at the early stages of their careers understand and apply cosmological parameter estimation techniques in modern cosmology.

These notebooks are designed to be used for constraining cosmological models, especially models such as (ΛCDM, oΛCDM, $w$CDM, and $w_0 w_a$CDM), that can be written in analytical form. The major focus of these notebooks is Bayesian inference through nested sampling using the PyPolyChord Python library. These notebooks are intended to help users understand not only how to run cosmological analyses, but also how to correctly construct likelihoods, apply proper statistical methods, and interpret the statistical quality of cosmological models.

The notebooks provide practical demonstrations of how to compute and interpret important statistical quantities commonly used in cosmology, including:

• Bayesian evidence and Bayes factors in logarithmic space,
• minimum chi-square statistics ($\chi^2_{\min}$),
• goodness-of-fit tests,
• $p$-values,

The first step is to install PyPolyChord on the local system. Here, I will provide installation examples for both Linux-based systems and macOS. For macOS, I expect the architecture to be arm64. However, even if the system is using an x86_64 architecture, one can still follow the Linux-based Miniconda installation procedure.

First, the computer needs to install essential libraries and compilers.

1. Ubuntu

Install Compiler

sudo apt update && sudo apt upgrade
sudo apt install nano
sudo apt install wget
sudo apt install git -y
sudo apt install liblapack-dev
sudo apt install libcfitsio-dev
sudo apt install build-essential
sudo apt-get install openmpi-bin openmpi-doc libopenmpi-dev

Install Python and Librareis, We recommd you to install Miniconda for better manage Python evironment.

mkdir -p ~/miniconda3
wget https://repo.anaconda.com/miniconda/Miniconda3-latest-Linux-x86_64.sh -O ~/miniconda3/miniconda.sh
bash ~/miniconda3/miniconda.sh -b -u -p ~/miniconda3
rm ~/miniconda3/miniconda.sh
source ~/miniconda3/bin/activate

2. MacOS

For Apple Silicon chip

mkdir -p ~/miniconda3
curl https://repo.anaconda.com/miniconda/Miniconda3-latest-MacOSX-arm64.sh -o ~/miniconda3/miniconda.sh
bash ~/miniconda3/miniconda.sh -b -u -p ~/miniconda3
rm ~/miniconda3/miniconda.sh
source ~/miniconda3/bin/activate

For Intel chip

mkdir -p ~/miniconda3
curl https://repo.anaconda.com/miniconda/Miniconda3-latest-MacOSX-x86_64.sh -o ~/miniconda3/miniconda.sh
bash ~/miniconda3/miniconda.sh -b -u -p ~/miniconda3
rm ~/miniconda3/miniconda.sh
source ~/miniconda3/bin/activate

From now on, I will give an example for Intel-based chips, where the architecture is x86_64. To install PyPolyChord, we need to do the following:

3. macOS when the architecture is x86_64

conda create --name polychord_env python=3.10 --platform osx-64
conda activate polychord_env
conda install -c conda-forge compilers openmpi mpi4py cmake make
git clone https://github.com/PolyChord/PolyChordLite.git
cd PolyChordLite
make
pip install .

4. macOS when the architecture is arm64

conda create -n cobaya_env python=3.10 -y
conda activate polychord_env
conda install -c conda-forge compilers openmpi mpi4py cmake make
git clone https://github.com/PolyChord/PolyChordLite.git
cd PolyChordLite
make
pip install .

Below, you will find links to several practical notebooks, which will help users understand how to construct likelihoods for different datasets such as Cosmic Chronometers, Pantheon$^+$ calibrated, Pantheon$^+$ uncalibrated, DES-Dovekie, Union3, and baryon acoustic oscillation (BAO) measurements from DESI Data Release 2 (DR2) considering cosmological models such as $\Lambda$CDM, o$\Lambda$CDM, $w$CDM, and $w_0 w_a$CDM. Each notebook is designed to help users understand the implementation of cosmological parameter estimation, including all relevant equations and mathematical arguments, likelihood construction, Bayesian inference, and statistical analysis using PyPolyChord. Keep in mind that, in these notebooks, I keep the sound horizon as a free parameter.

Keep in mind that, in the three supernova notebooks above, I marginalize over the nuisance parameter $\mathcal{M}$. However, in the notebook below, I instead vary $\mathcal{M}$ as a free parameter