On the True Significance of the Hubble Tension: A Bayesian Error Decomposition Accounting for Information Loss

The Hubble tension, a persistent discrepancy between early and late Universe measurements of $H_0$, poses a significant challenge to the standard cosmological model. In this work, we present a new Bayesian hierarchical framework designed to meticulously decompose this observed tension into its constituent parts: standard measurement errors, information loss arising from parameter-space projection, and genuine physical tension. Our approach, employing Fisher matrix analysis with MCMC-estimated loss coefficients and explicitly modeling information loss via variance inflation factors ($\lambda$), is particularly important in high-precision analysis where even seemingly small information losses can impact conclusions. We find that the real tension component ($T_{real}$) has a mean value of 5.94 km/s/Mpc (95% CI: [3.32, 8.64] km/s/Mpc). Quantitatively, approximately 78% of the observed tension variance is attributed to real tension, 13% to measurement error, and 9% to information loss. Despite this, our decomposition indicates that the observed $\sim$$6.39\sigma$ discrepancy is predominantly a real physical phenomenon, with real tension contributing $\sim$$5.64\sigma$. Our findings strongly suggest that the Hubble tension is robust and probably points toward new physics beyond the $\mathrm{\Lambda }$CDM model.