### Bayesian Analysis Chains for the NANOGrav 12.5 Year Data Set

As used in The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background.

These files are the output from a Markov chain Monte Carlo (MCMC) sampling process for various models. Each chain file has many rows, each of which gives a set of possible values for the model parameters. The chance of any given set of parameter values appearing in the file is proportional to the posterior probability for those parameter values: the product of the likelihood of the observed times of arrival given those parameter values and the prior probability of those parameter values.

The values in the chain file are separated by tab characters. The parameter names are given in the associated parameters file. The parameters are the amplitude A and spectral index γ of the intrinsic red noise for each pulsar, followed by the parameters specific to the common red noise (gravitational wave background) model. Each row of the chain file has 4 additional meta-parameters:

- Natural log of posterior probability
- Natural log of likelihood for this parameter set
- Average jump acceptance of the MCMC process for this and previous samples
- Parallel tempering swap acceptance, always 1.0 because parallel tempering was not used in these analyses.

The first rows of each chain file come from the beginning of the MCMC process, during the "burn in" process. These samples are not representative of the true posterior probability and should not be used. We recommend skipping the first 30000 rows when using these chains.

When we model a background spectrum, we consider a set of Gaussian-distributed amplitudes for sine and cosine modes of frequencies n/T, where T is the total time of observation, about 12.88 years, and n = 1..5 or 1..30 as given below. Click here for the exact frequencies in Hz.

Our models are the following:

- Power law spectrum with spectral index fixed to 4.33: parameter is the amplitude.
- Power law with varying spectral index: parameters are the index γ and the amplitude.
- Free spectrum: parameters are the powers in the first 30 Fourier modes.

The interpretation of times of arrival depends on the solar system ephemeris. Here we have generally used the DE438 ephemeris but sometimes the INPOP19 ephemeris.

Below we give either the chains and the parameter names as two separate files or the two together in an HDF5 file. Detailed examples of how to use this data are given in the Tutorials. We also reproduce there the figures of our 12.5yr stochastic background paper.

Here are the files:

- Runs using the DE438 ephemeris:
- Fixed spectral index, 5 frequencies: Chains + Parameters | HDF5
- Varying spectral index, 5 frequencies: Chains + Parameters | HDF5
- Varying spectral index, 30 frequencies: Chains + Parameters | HDF5
- Broken power law, 30 frequencies: Chains + Parameters | HDF5
- Free spectrum, 30 frequencies: Chains + Parameters | HDF5
- Factorized likelihood: Tarball. Factorized likelihood runs are likelihoods for common red noise processes for the pulsars taken one at a time. By multiplying the individual pulsar likelihoods together, we can find the common red noise likelihoods of the combined pulsars. See section 5.1 of the 12.5 year stochastic background paper.

- Runs using DE438 using
BayesEphem,
which varies parameters of the ephemerides that modify the
solar system barycenter as parameters of our model:
- Fixed spectral index, 5 frequencies: Chains + Parameters | HDF5

- Run using the INPOP19 ephemeris:
- Fixed spectral index, 5 frequencies: Chains + Parameters | HDF5

- Run using INPOP19 ephemeris with BayesEphem:
- Fixed spectral index, 5 frequencies: Chains + Parameters | HDF5