In many practical situations parameters are only weakly constrained by data. Prior physical knowledge can therefore be incorporated using Bayesian regularization.

Each selected parameter $p_j$ is assumed to follow a Gaussian prior

$$ p_j \sim \mathcal{N}(\mu_j,\sigma_{p_j}), $$

with prior mean $μ_j$ and prior width $\sigma_{p_j}$ introducing a Bayesian penalty term

$$ \chi^2_{\mathrm{Bayes}} = \sum_j \frac{(p_j-\mu_j)^2}{\sigma_{p_j}^2}. $$

The minimized objective becomes

$$ \chi^2_{\mathrm{total}} = \chi^2_{\mathrm{data}} + \chi^2_{\mathrm{Bayes}}, $$

which corresponds to Maximum A Posteriori (MAP) estimation.

Bayesian terms act as soft parameter constraints rather than fixed limits.

Global Fit Formulation

For $K$ datasets with $N_k$ points:

$$ \chi^2_{\mathrm{data}} = \sum_{k=1}^{K} \sum_{i=1}^{N_k} r_{k,i}^2, $$

with residuals

$$ r_{k,i} = \frac{ I_{\mathrm{calc},k}(q_{k,i}) - I_{\mathrm{exp},k}(q_{k,i}) }{\sigma_{k,i}}. $$

Total number of residuals:

$$ N=\sum_k N_k. $$

Global parameters influence all datasets simultaneously, while local parameters affect only individual datasets.

Optimization Algorithms

Two complementary minimization strategies are used.

Simplex Method

The Nelder–Mead simplex algorithm minimizes directly

$$ \chi^2_{\mathrm{total}}(\mathbf{p}). $$

Advantages:

• derivative–free,
• robust far from minimum,
• suitable for initialization.

However convergence close to the optimum is relatively slow.

Levenberg–Marquardt Method

The Levenberg–Marquardt (LM) algorithm minimizes

$$ \chi^2=\sum_i f_i^2 $$

using a residual vector $f_i$ and Jacobian matrix

$$ J_{ij}=\frac{\partial f_i}{\partial p_j}. $$

LM combines gradient descent and Gauss–Newton methods:

$$ (J^T J + \lambda I)\Delta p = -J^T f. $$

It provides fast quadratic convergence near the solution and enables reliable error estimation.

Bayesian Residual Formulation for LM

Standard LM implementations (e.g. GNU Scientific Library) require the number of residuals to remain equal to the number of experimental data points.

Instead of adding extra Bayesian residuals, the Bayesian contribution is distributed over all data residuals.

First, the normalized Bayesian term is defined as

$$ B = \frac{1}{N} \sum_j \frac{(p_j-\mu_j)^2}{\sigma_{p_j}^2}. $$

Each residual is then modified as

$$ f_i = \mathrm{sign}(r_i) \sqrt{r_i^2 + B}. $$

This formulation guarantees

$$ \sum_i f_i^2 = \chi^2_{\mathrm{data}} + \chi^2_{\mathrm{Bayes}}, $$

while preserving the required residual dimension.

The Bayesian information therefore influences every data point equally during minimization.

Jacobian Matrix

The LM algorithm requires derivatives of the modified residuals:

$$ J_{ij} = \frac{\partial f_i}{\partial p_j}. $$

Using the residual definition above,

$$ \frac{\partial f_i}{\partial p_j} = \frac{ r_i \frac{\partial r_i}{\partial p_j} + \frac12 \frac{\partial B}{\partial p_j} }{ \sqrt{r_i^2+B} }. $$

Model contribution:

$$ \frac{\partial r_i}{\partial p_j} = \frac{1}{\sigma_i} \frac{\partial I_{\mathrm{calc}}}{\partial p_j}. $$

Bayesian contribution:

$$ \frac{\partial B}{\partial p_j} = \frac{2}{N} \frac{p_j-\mu_j}{\sigma_{p_j}^2}. $$

Thus Bayesian priors modify all Jacobian rows without increasing matrix size.

Relation to Regularization

The Bayesian penalty is mathematically equivalent to Tikhonov regularization:

$$ \chi^2 = ||r||^2 + ||L(\mathbf{p}-\mathbf{\mu})||^2. $$

Bayesian fitting therefore stabilizes ill-posed problems, reduces parameter correlations, and suppresses nonphysical solutions.

Infinite prior width reproduces classical least squares.

Parameter Errors

After convergence the covariance matrix is obtained from

$$ C = (J^T J)^{-1}. $$

Parameter uncertainties are

$$ \sigma_{p_j}=\sqrt{C_{jj}}. $$

Bayesian priors effectively add statistical information, leading to finite and stable error estimates even for weakly determined parameters.

Practical Workflow

Typical fitting procedure:

1. Simplex global search
2. Levenberg–Marquardt refinement
3. covariance evaluation
4. uncertainty estimation

This combined Bayesian global fitting approach provides robust convergence and statistically consistent parameter errors for complex nonlinear models.

• Global/Local Limits: instead of the range select a bayesian option: instead min/max put prior mean value and prior width