Fitting in a complex chi^2 landscape using an optimized hypersurface sampling

Fitting a data set with a parametrized model can be seen geometrically as finding the global minimum of the chi^2 hypersurface, depending on a set of parameters {P_i}. This is usually done using the Levenberg-Marquardt algorithm. The main drawback of this algorithm is that despite of its fast convergence, it can get stuck if the parameters are not initialized close to the final solution. We propose a modification of the Metropolis algorithm introducing a parameter step tuning that optimizes the sampling of parameter space. The ability of the parameter tuning algorithm together with simulated annealing to find the global chi^2 hypersurface minimum, jumping across chi^2{P_i} barriers when necessary, is demonstrated with synthetic functions and with real data.

💡 Research Summary

The paper addresses the well‑known difficulty of fitting experimental data with a parametrized model when the χ² surface is rugged and contains many local minima. Traditional deterministic algorithms such as Levenberg‑Marquardt (LM) converge quickly only if the initial guess lies close to the global minimum; otherwise they become trapped in sub‑optimal basins. To overcome this limitation, the authors adopt a Bayesian framework and employ a Metropolis‑Hastings Markov Chain Monte Carlo (MCMC) sampler, which allows uphill moves in χ² space according to a probabilistic acceptance rule.

A central contribution is an adaptive scheme for the proposal step sizes (ΔP_max_i) associated with each model parameter. After a fixed number of MCMC steps (a “regeneration” interval), the algorithm computes the actual acceptance ratio R_i for each parameter and compares it with a desired ratio R_i,desired = R_desired / m (where m is the number of parameters). The step size is then updated via a Robbins‑Monro‑type rule: ΔP_new_i = ΔP_old_i × (R_i / R_i,desired). This feedback loop automatically enlarges a step if the corresponding parameter is accepted too often (indicating over‑exploration) and shrinks it if acceptance is too rare (indicating under‑exploration). Consequently, all parameters evolve with roughly the same acceptance probability, which improves sampling efficiency in high‑dimensional spaces.

In addition to step‑size adaptation, the authors integrate simulated annealing. An artificial temperature T is introduced into the Metropolis acceptance probability, exp