Likelihood

We just looked at how the scipy.optimize.curve_fit function can be used to obtain the optimal parameters the agreement between some model and data. To achieve this, the curve_fit function is maximising the likelihood of the model to the data, \(\mathcal{L}\big[\mathbf{D} | M(\mathbf{X})\big]\). For the specific case of Gaussian uncertainties, this may be called minimising the \(\chi^2\)-statistic.

The likelihood, given the model \(M\) and a set of parameters \(\mathbf{X}\), of the data \(\mathbf{D}\) where these data are assumed to have Gaussian uncertainties can be found as,

\[ \ln\mathcal{L}\big[\mathbf{D} | M(\mathbf{X})\big] = -\frac{1}{2} \sum_n\Bigg({\bigg[\frac{y_n - M(\mathbf{X})}{\sigma_n}\bigg]^2 + \ln(2\pi\sigma_n)\Bigg)}, \]

where, \(y_n\) is the \(n\)-th data value and \(\sigma_n\) the associated uncertainty.

The likelihood is a probability distribution, that represents how well the parameter values reproduce the experimental data. In the figure below, we show how the likelihood is affected by a change in \(A\) and \(\gamma\) for the Lorentzian modelling of QENS data discussed previously. Note that the values of \(x_0\) and \(b\) are constrained to the optimal solutions found previously while \(A\) and \(\gamma\) were varied, the code to generate this figure can be found here.

An example of a likelihood function, where $A$ and $\gamma$ are varied.

It is clear that the likelihood is maximised at the optimised solutions for \(A\) and \(\gamma\) found previously, and indeed the shape of the likelihood appears to be Gaussian in nature. These aspects will become important later in the course.

Model-dependent analysis Classification