Analysis of KATRIN data using Bayesian inference

The KATRIN (KArlsruhe TRItium Neutrino) experiment will be analyzing the tritium beta-spectrum to determine the mass of the neutrino with a sensitivity of 0.2 eV (90% C.L.). This approach to a measurement of the absolute value of the neutrino mass relies only on the principle of energy conservation and can in some sense be called model-independent as compared to cosmology and neutrino-less double beta decay. However by model independent we only mean in case of the minimal extension of the standard model. One should therefore also analyse the data for non-standard couplings to e.g. righthanded or sterile neutrinos. As an alternative to the frequentist minimization methods used in the analysis of the earlier experiments in Mainz and Troitsk we have been investigating Markov Chain Monte Carlo (MCMC) methods which are very well suited for probing multi-parameter spaces. We found that implementing the KATRIN chi squared function in the COSMOMC package - an MCMC code using Bayesian parameter inference - solved the task at hand very nicely.

💡 Research Summary

The paper presents a novel statistical approach for analyzing the tritium beta‑decay spectrum measured by the KATRIN (Karlsruhe Tritium Neutrino) experiment, which aims to determine the absolute electron‑neutrino mass with a target sensitivity of 0.2 eV (90 % confidence level). Historically, the analysis of KATRIN‑type data has followed the frequentist paradigm: a χ² function is constructed from the theoretical spectrum (including signal amplitude, background rate, endpoint energy, and neutrino‑mass‑squared) and then minimized using the Minuit2 package. While this method works well for the four‑parameter “standard” model, it suffers from several drawbacks when the parameter space is enlarged or when the χ² surface contains multiple minima. The authors therefore explore a Bayesian alternative based on Markov Chain Monte Carlo (MCMC) sampling, implemented through the publicly available COSMOMC code, originally designed for cosmological parameter estimation.

The authors first recast the KATRIN χ² function into a likelihood L = exp(−χ²/2) and adopt flat (uninformative) priors for all parameters. They then embed this likelihood into COSMOMC, disabling the cosmology modules and defining four free parameters: the neutrino‑mass‑squared (m²νe), the endpoint energy (E₀), the signal amplitude (A), and the background rate (B). After generating synthetic spectra with known input values (m²νe = 0 eV², E₀ = 18575 eV, A ≈ 477.5 Hz, B = 0.01 Hz) they run multiple MCMC chains, discard a 50 % burn‑in, and monitor convergence using the Gelman‑Rubin R‑statistic (requiring 1 − R < 0.03). The resulting posterior distributions are visualized with GetDist, providing both one‑dimensional marginalized histograms and two‑dimensional contour plots.

Key results for the minimal model are:

• m²νe = −4.1 × 10⁻⁶ ± 1.3 × 10⁻² eV²,
• ΔE₀ = 8.7 × 10⁻⁶ ± 2.2 × 10⁻³ eV,
• B = 1.00 × 10⁻² ± 1.5 × 10⁻⁴ Hz,
• A = 477.0 ± 0.16 Hz.

The statistical uncertainty on m²νe (0.013 eV²) is about 23 % smaller than the uncertainty obtained from a traditional frequentist analysis of the same simulated data set (≈0.016 eV²). Moreover, the two‑dimensional posterior plots reveal strong correlations, especially between (A, B) and (E₀, m²νe), which are difficult to quantify with the standard covariance‑matrix approach. These correlations are crucial for realistic error budgeting and for guiding experimental design choices such as the allocation of measurement time across the retarding‑potential spectrum.

Beyond the minimal four‑parameter case, the Bayesian framework offers a natural path to incorporate non‑standard physics, such as right‑handed currents or sterile‑neutrino admixtures, which would introduce additional parameters and potentially complex χ² landscapes. In a frequentist setting, adding such parameters often leads to poor convergence, multiple local minima, and prohibitive computational cost. In contrast, MCMC explores the entire parameter space, can jump between separated modes, and provides a full posterior probability distribution for each new parameter, allowing straightforward marginalization and credible‑interval extraction.

The authors also discuss practical aspects of the implementation: the need to choose appropriate step sizes, the importance of a sufficiently long burn‑in, and the use of convergence diagnostics (R‑statistic, visual inspection of trace plots). They note that while MCMC requires more computational time per spectrum than a single Minuit2 minimization, the overall workflow scales favorably when many spectra or a full theoretical model are analyzed, because the posterior sampling yields all statistical information in one run.

In conclusion, the paper demonstrates that embedding the KATRIN χ² function into COSMOMC and performing Bayesian inference via MCMC yields several advantages: (1) robust global exploration of multi‑dimensional parameter spaces, (2) explicit quantification of parameter correlations, (3) modest improvement in statistical precision for the neutrino‑mass parameter, and (4) a flexible platform for extending the analysis to beyond‑standard‑model scenarios. These methodological gains are expected to be valuable when KATRIN begins taking real data, especially for searches of subtle non‑standard signatures that would otherwise be obscured by the limitations of traditional frequentist techniques.