I have a sample consisting of $N$ data for the function

$$g(\nu) = \int\limits_0^{\infty} \cos{\left(\nu x\right)} f(x) \,\mathrm d x$$

and I want to solve the integral equation for $f(x)$ numerically.

Recently, I come across a numerical solution that uses Bayesian inference. Essentially, in order to solve the problem, one includes some additional information, i.e., the default model $m(x)$ that encodes the prior information we have about $f(x)$. Then, the solution can be found by maximising the quantity

$P(f|g,m)\propto\exp{\left(-\frac{1}{2}\sum\limits_{i,j=0}^N\left[g_i-\sum_{l=0}^{M}\cos{(\nu_i \Delta x \,l)}f_l \right]\left(\Sigma^{-1}\right)_{ij}\left[g_j-\sum_{l=0}^{M}\cos{(\nu_j \Delta x \,l)}f_l\right]-\sum\limits_{l=0}^M\Delta x\left[f_l-m_l-f_l\ln{\left(\frac{f_l}{m_l}\right)}\right]\right)}$

with respect to $f_l\equiv f(\Delta x \, l)$ for $l\in[0,M]$. In the previous equation I've defined $g_i\equiv g(\nu_i)$ (the available data), where $\nu_i$ for $i \in [0,N]$. Moreover $\Sigma$ is the covariance matrix of data $g_i$.
Does anyone has some idea on how to face this minimisation problem numerically?



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