Systematic errors

Since neither the physical base model nor the surrogate happens to perfectly represent the physics of the transient, the likelihood function in fiesta uses a systematic uncertainty \(\sigma_{\rm sys}\).

\[\mathcal{L}(\vec{\theta}|d) = - \frac{1}{2} \sum_{t_j} \biggl( \frac{(m(t_j) - m^{\star}(t_j, \vec{\!\theta}\,))^2}{\sigma(t_j)^2 + \sigma_{\text{sys}}^2} + \ln(2\pi (\sigma(t_j)^2 + \sigma_{\text{sys}}^2)) \biggr)\]

In fiesta, this is controlled through the EMLikelihood class. There are three ways \(\sigma_{\rm sys}\) can be set up (see below). Which one of these modes is chosen depends on which _setup_sys_uncertainty_* method is called on the EMLikelihood instance. Upon initialization, the likelihood instance is in the “fixed” systematic uncertainty mode. When doing inference, this is determined automatically depending on whether a systamics .yaml file is provided and whether the prior includes the sys_err variable. The three modes are:

Fixed

In this case, \(\sigma_{\rm sys}\) is just a constant determined through the error_budget keyword when initializing the likelihood function (defaults to 0.3 mag). In the inference this happens, when neither a systematics file is provided in the Fiesta sampler instance and the prior does not contain a sys_err parameter.

Free

In this case, \(\sigma_{\rm sys}\) is the same across all filters and times, but is sampled freely from a prior. This prior needs to be specified as sys_err in the prior. When initializing a Fiesta sampler with such a prior, the likelihood function is updated accordingly, hence its value for error_budget will be overwritten.

Time-dependent

In this case, the systematic uncertainty is sampled freely, is time-depdendent, and can also vary between filters. For this to happen, the Fiesta sampler needs to be initialized with the argument systematics_file, where systematics_file points to a .yaml file that will determine the specific setup. This .yaml config should provide time_nodes, i.e. the number of time nodes for which the systematic uncertainties are sampled. It should also specify a fiesta.inference.prior type and its parameters, except for the naming. For instance, to introduce four sampling parameters that represent the systematic uncertainty for all filters collectively, using four time nodes linearly spaced between 0.3 and 26 days, one could set up:

collective:
    time_nodes: 4
    time_range: linear 0.3 26
    prior: Uniform
    params:
           xmin: 0.3
           xmax: 1.0

Alternatively, if each filter should have their own systematic uncertainty parameters, the following works (here we omit the time_range argument, which causes the nodes to be spaced linearly between EMLikelihood.tmin and EMLikelihood.tmax):

individual:
    time_nodes: 4
    prior: Uniform
    params:
           xmin: 0.3
           xmax: 1.0

We can also group several filters together:

group1:
    filters:
        - "X-ray-5keV"
        - "X-ray-1keV"
    time_nodes: 4
    time_range: linear 0.1 10
    prior: Uniform
    params:
           xmin: 0.3
           xmax: 2

group2:
    filters:
        - "radio-3GHz"
        - "radio-6GHz"
    time_nodes: 2
    time_range: log 0.3 1000
    prior: LogUniform
    params:
           xmin: 0.3
           xmax: 1.0

remaining:
    time_nodes: 4
    prior: Uniform
    params:
           xmin: 0.3
           xmax: 1.0

This would mean the X-ray filters share their 4 systematic uncertainty parameters spaced linearly between 0.1 and 10 days, the radio filters are sampled with two separate systematic uncertainty filters spaced geometrically between 0.3 and 1000 days, and all remaining filters are sampled with a different set of four systematic uncertainty parameters that are linearly spaced between between EMLikelihood.tmin and EMLikelihood.tmax. If the data does not contain any remaining filters except X-ray-5keV, X-ray-1keV, radio-3GHz, radio-6GHz, then remaining is ignored.