A comparative Bayesian PINN–MCMC analysis of Barrow–Tsallis holographic dark energy with neutrinos: Toward resolving the Hubble tension

We present a comprehensive analysis of the Barrow–Tsallis Holographic Dark Energy (BTHDE) model within a spatially flat FLRW cosmology, focusing on a comparative assessment between traditional Markov Chain Monte Carlo (MCMC) methods and a Bayesian Physics-Informed Neural Network (PINN). By embedding the modified Friedmann dynamics directly into the learning architecture, the Bayesian PINN framework enforces physical consistency while enabling efficient posterior sampling. We employ a broad suite of cosmological observations, including Planck 2018 CMB temperature and polarization data, CMB lensing, Baryon Acoustic Oscillations (BAO), Cosmic Chronometers (CC), and the Pantheon+ Type Ia supernovae compilation, to jointly constrain the Hubble constant H 0 , the Tsallis nonextensive parameter q , Barrow exponent Δ, Granda–Oliveros cutoff parameters α and β , and the total neutrino mass Σ m ν . In the late-time (CC + Pantheon+) analyses, we adopt flat, physically motivated priors on the baryon and cold dark matter densities ( Ω b h 2 and Ω c h 2 ) solely for the internal calibration of the sound horizon r d , without employing any Planck-based or early-universe likelihoods. The Bayesian PINN consistently yields sharper posterior distributions than MCMC, particularly for H 0 and Σ m ν , while ensuring solution smoothness and dynamical compatibility. Both methods place the inferred H 0 values between the Planck and SH 0 ES estimates, lowering the Hubble tension to the 1.3 σ –2.1 σ range. In the MCMC framework, combined CMB and low-redshift datasets constrain Σ m ν < 0.114 eV and H 0 = 70.6 ± 1.35 km/s/Mpc . Our results demonstrate the viability of the BTHDE model as a compelling alternative to ΛCDM and highlight the complementary strengths of Bayesian PINNs and MCMC in probing extended dark energy scenarios governed by generalized entropy frameworks.