Derivatives w.r.t. noise parameters for derivative-aware MCMC methods

[abhisrkckl]

The timing model parameters implement two types of derivatives - d_delay_d_PARAM or d_phase_d_PARAM depending on whether they enter the timing model via a delay or a phase correction. These types of derivatives are not appropriate for noise parameters as the timing residuals do not depend on them.

To implement a derivative-aware MCMC method (#1303) such as HMC, we need derivatives of the form d_lnlikelihood_d_PARAM, where lnlikelihood contains both the "chi-squared" term and the normalization term. For timing model parameters it is easy to get these derivatives using the design matrix, which in turn is computed using the d_delay_d_PARAM and d_phase_d_PARAM functions.

We should implement derivative functions so that we can compute d_lnlikelihood_d_PARAM for noise parameters. These parameters enter the lnlikelihood function via the covariance matrix, which is computed as
C = N + F^T Φ F
where N is the diagonal white noise covariance matrix (containing scaled toa uncertainties), F is the correlated noise basis and Φ is the diagonal correlated noise weight matrix.

I propose we implement the following:

• For white noise parameters (EFAC and EQUAD) - d_scaledtoaerrs_d_PARAM
• For correlated noise parameters - d_noiseweights_d_PARAM and d_noisebasis_d_PARAM

For most correlated noise parameters only d_noiseweights_d_PARAM will be present as the noise basis is independent of the model parameters (see #1363). d_noisebasis_d_PARAM will be needed only in rare cases, such as as-yet-unimplemented TNREDFLOW parameter (fundamental frequency of the red noise basis).