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Likelihood Functions

This page defines the function that the sampler evaluates for each parameter vector: a Gaussian likelihood on 1,590 supernova distance moduli and 13 BAO ratios, each with its full covariance, multiplied by uniform priors.

Combined likelihood​

ln⁡p(θ∣data)=ln⁡prior(θ)−12[χSN2(θ)+χBAO2(θ)]\ln p(\theta \mid \text{data}) = \ln \text{prior}(\theta) - \tfrac{1}{2}\left[\chi^2_{\mathrm{SN}}(\theta) + \chi^2_{\mathrm{BAO}}(\theta)\right] χSN2=ΔμT CSN−1 Δμ,Δμ=μobs−μth(z;θ)\chi^2_{\mathrm{SN}} = \Delta\mu^{\mathsf T}\, C_{\mathrm{SN}}^{-1}\, \Delta\mu, \qquad \Delta\mu = \mu_{\mathrm{obs}} - \mu_{\mathrm{th}}(z;\theta) χBAO2=ΔBT CBAO−1 ΔB,ΔB=Bobs−Bth(z;θ)\chi^2_{\mathrm{BAO}} = \Delta B^{\mathsf T}\, C_{\mathrm{BAO}}^{-1}\, \Delta B, \qquad \Delta B = B_{\mathrm{obs}} - B_{\mathrm{th}}(z;\theta)
  • ln prior is 0 inside the bounds and −∞ outside (top-hat priors; see Prior Specification).
  • The covariance matrices are fixed, so the determinant term of the Gaussian is a constant and is dropped.
  • The two probes are summed as independent (Covariance Handling).
  • There is no nuisance parameter (no absolute-magnitude offset): the only parameters are those of the model.

Model predictions​

QuantityDefinition
Luminosity distanceD_L = (1+z) D_C, with D_C = (c/H₀) ∫₀ᶻ dz′/E(z′)
Supernova predictionμ_th = 5 log₁₀(D_L / Mpc) + 25
BAO predictionsD_M/r_d, D_H/r_d, D_V/r_d with D_H = c/(H₀E), D_V = [z D_M² D_H]^(1/3), rd = 147.09 Mpc

with c = 299792.458 km/s, flat geometry (DM = DC) and E(z) from the model.

Numerical integration​

distance_grids evaluates E(z) on a uniform grid of 3,000 points from 0 to the largest redshift of the probe being evaluated, integrates 1/E with a cumulative trapezoidal rule, and linearly interpolates DL, DC and DH at the data redshifts. The grid is rebuilt on every call, separately for supernovae and for BAO.

Numerical accuracy is not validated in the repository

The tests of the repository check the result tables and the AIC/BIC arithmetic, not the distance integrals. The grid resolution is a fixed choice, not a convergence-tested one.

Sampling target​

log_probability(params, model, data, xi_helper) returns ln prior − ½ χ², or −∞ as soon as a parameter leaves its bounds. Because the prior is flat, the log-probability of a sample is −½χ² up to a constant: this is what makes the "best sampled point" a usable χ² estimate (Model Comparison).