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.