False Positive Probability Calculation

vespa calculates the false positive probability for a transit signal as follows:

\[{\rm FPP} = 1 - P_{\rm pl},\]

where

\[P_{\rm pl} = \frac{\mathcal L_{\rm pl} \pi_{\rm pl}} {\mathcal L_{\rm pl} \pi_{\rm pl} + \mathcal L_{\rm FP} \pi_{\rm FP}}.\]

The \(\mathcal L_i\) here represent the “model likelihood” factors and the \(\pi_i\) represent the “model priors,” with the \({\rm FP}\) subscript representing the sum of \(\mathcal L_i \pi_i\) for each of the false positive scenarios.

Likelihoods

Each EclipsePopulation contains a large number of simulated instances of the particular physical scenario, each of which has a simulated eclipse shape and a corresponding trapezoidal fit. This enables each population to define a 3-dimensional probability distribution function (PDF) for these trapezoid parameters, \(p_{\rm mod} (\log_{10} (\delta), T, T/\tau)\). As the TransitSignal object provides an MCMC sampling of the trapezoid parameters for the observed transit signal, the likelihood of the transit signal under a given model can thus be approximated as a sum over the model PDF evaluated at the \(K\) samples:

\[\mathcal L = \displaystyle \sum_{k=1}^K p_{\rm mod} \left(\log_{10} (\delta_k), T_k, (T/\tau)_k\right)\]

This is implemented in EclipsePopulation.lhood().

Priors

Each EclipsePopulation also has a EclipsePopulation.prior attribute, the value of which represents the probability of that particular astrophysical scenario existing. For a BEBPopulation, for example, the prior is (star density) * (sky area) * (binary fraction) * (eclipse probability). If observational constraints are applied to a population, then an additional selectfrac factor will be multiplied into the prior, representing the fraction of scenarios that are still allowed to exist, given the constraints.