tokens: timing of the first transit. As such, the mean anomaly of 159c was different for each simulation but was not a free parameter. This leaves eight unknown parameters, plus seven for the presumed third planet. This is a 15-dimensional parameter space to search. To further reduce computational requirements, we reduced this to 11 parameters. This reduction was accomplished by presuming 159b was in a circular orbit, using its orbital plane as the reference for the system, and using a constant estimate of its mass. Thus, 159b’s eccentricity, argument of periastron, longitude of the ascending node, and mass are removed from the priors. This is discussed in more detail in Section 3.4.2 . 3.2 TTVFast and MultiNest We used TTVFast (Deck et al. 2014 ), a symplectic numerical integrator, to compute the transit times for each hypothesized set of orbital parameters. These times were converted to TTVs, then compared to the observed TTV data from Holczer et al. ( 2016 ). Another set of parameters would then be chosen by MultiNest, and the values compared again. To search the parameter space efficiently, we used the Bayesian Inference algorithm MultiNest (Feroz et al. 2009 , 2013 ). This iterative process can find solutions far more quickly than searching the entire parameter space. We used the Python interface for MultiNest, PyMultiNest by Buchner et al. ( 2014 ) for this project. Due to the size and complexity of the parameter space, finding the best fits required an iterative process and dozens of sets of runs. Each ’run’ produces a single result from MultiNest, estimated from millions of samples of the parameter space and as many simulations from TTVFast. Repeated runs, even with the same priors, would often return different best-fits. This is indicative of multiple maxima in our parameter space, particularly associated with different resonances between 159d and 159c. It is known that resonances produce strong TTVs and that there can be degeneracy between resonances in this regard (Boué et al. 2012 ). To examine this effect, we performed additional runs with periods of 159d constrained to be near the resonances (periods within 10 days of resonance) of 159c. This narrowing of ranges allowed us to determine the best-fits associated with each resonance independent of the others. We used the default parameters of PyMultiNest with the exception of the evidence tolerance, which we set to 0.15 (from default of 0.5). This reduced tolerance value allows for a more fine-tuned search across the peaks of this rapidly varying landscape. 3.3 Likelihood The Likelihood, L we used is based on the usual χ 2 statistic and is given by: where x i is the calculated TTV, μ i is the observed TTV, and σ i is the error in the observed TTV for point i , respectively (Ivezic et al. 2014 ). The log-likelihood is the value we aim to maximize in the simulation In this case, summation includes transit data from both 159b and 159c. 3.4 Priors The priors used for the search were constrained
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