Maximum-likelihood inversion

For a given measured friction curve (\(\mu_{obs}\)), it is likely that a single set of parameters \(\mathcal{P}\) produces the best possible agreement between a forward simulation \(\mu = \mathrm{Forward}[t, \mathcal{P}]\) and the (noisy) measurements. This single optimal estimate of \(\mathcal{P}\) is called the maximum-likelihood estimate, and it can be obtained through standard inversion schemes.

For the least-squares objective function discussed in the previous section, the Levenberg-Marquardt algorithm is one of the fastest converging methods on the market. DiaBayes utilises the LevenbergMarquardt implementation of the Optimistix library, which interfaces directly with the diffrax.diffeqsolve routine that is used to solve the forward problem.

Concretely, performing a maximum-likelihood inversion in DiaBayes is fairly straightforward. Assuming that a solver has already been instantiated with an appropriate forward model (see the Getting Started guide), and given some measured friction curve mu_measured and initial guess parameters params_guess, the inversion can be performed with a single function call:

inv_result = solver.max_likelihood_inversion(
    t, mu_measured,
    y0=y0, params=params_guess,
    friction_constants=constants,
    block_constants=block_constants,
    verbose=True
)

params_inverted = inv_result.value

Depending on the number of data samples and the complexity of the forward model, this inversion should take anywhere from a few seconds to at most one minute.

In many cases, the maximum-likelihood result is sufficient for the analysis of an experiment. However, this approach provides no information on the uncertainty in the inverted parameters, and the potential trade-offs between them. For a complete evaluation of the admissible range of parameters, consider performing a Bayesian inversion (hint: see next section).