Priors
All parameters to your model must have a prior defined. You may provide any continuous, univariate distribution from the Distributions.jl. A few useful distributions include:
NormalUniformLogNormalLogUniformTrucatedNormalVonMises
This pacakge also defines the Sine() distribution for e.g. inclination priors and UniformCircular() for periodic variables. Internally, UniformCircular() creates two standard normal variables and finds the angle between them using arctan. This allows the sampler to smoothly cycle past the ends of the domain. You can specify a different circular domain than (0,2pi) by passing the size of the domain e.g. τ = UniformCircular(1.0).
The VonMise distribution is notable but not commonly used. It is the analog of a normal distribution defined on a circular domain (-π, +π). If you have a Gaussian prior on an angular parameter, a Von Mises distribution is probably more appropriate.
Behind the scenes, Octofitter remaps your parameters to unconstrained domains using the Bijectors.jl (and corrects the priors accordingly). This is essential for good sampling efficiency with HMC based samplers.
This means that e.g. if you define the eccentricity prior as e=Uniform(0,0.5), the sampler will actually generate values across the whole real line and transform them back into the [0,0.5] range before evaluating the orbit. It is therefore essential that your priors do not include invalid domains.
For example, setting a=Normal(3,2) will result in poor sampling efficiency as sometimes negative values for semi-major axis will be drawn (especially if you're using the parallel tempered sampler).
Instead, for parameters like semi-major axis, eccentricity, parallax, and masses, you should truncate any distributions that have negative tails. This can easily be accomplished with TrauncatedNormal or Trunacted(dist, low, high) for any arbitrary distribution.
Kernel Density Estimate Priors
Octofitter has support for sampling from smoothed kernel density estimate priors. These are non-parametric distributions fit to a 1D dataset consisting of random draws. This is one way to include the output of a different model as the prior to a new model. That said, it's usually best to try and incorporate the model directly into the code. There are a few examples on GitHub of this, including atmosphere model grids, cooling tracks, etc.
Using a KDE
First, we will generate some data. In the real world, you would load this data eg. from a CSV file.
using Octofitter, Distributions
# create a smoothed KDE estimate of the samples from a 10+-1 gaussian
kde = Octofitter.KDEDist(randn(1000).+10)KDEDist kernel density estimate distribution
Note that in Octofitter the KDE will have its support truncated to the minimum and maximum values that occur in your dataset, ie. it doesn't allow for infinite long tails.
Now add it to your model as a prior:
@planet b Visual{KepOrbit} begin
a ~ kde # Sample from the KDE here
e ~ Uniform(0.0, 0.99)
i ~ Sine()
ω ~ UniformCircular()
Ω ~ UniformCircular()
θ ~ UniformCircular()
tp = θ_at_epoch_to_tperi(system,b,50000)
end
@system Tutoria begin
M ~ truncated(Normal(1.2, 0.1), lower=0.1)
plx ~ truncated(Normal(50.0, 0.02), lower=0.1)
end b
model = Octofitter.LogDensityModel(Tutoria)
chain = octofit(model)Chains MCMC chain (1000×28×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 0.73 seconds
Compute duration = 0.73 seconds
parameters = M, plx, b_a, b_e, b_i, b_ωy, b_ωx, b_Ωy, b_Ωx, b_θy, b_θx, b_ω, b_Ω, b_θ, b_tp
internals = n_steps, is_accept, acceptance_rate, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt, loglike, logpost, tree_depth, numerical_error
Summary Statistics
parameters mean std mcse ess_bulk ess_tail r ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Floa ⋯
M 1.1979 0.1037 0.0031 1145.2492 482.4900 0.9 ⋯
plx 49.9996 0.0199 0.0006 1083.2175 712.4667 1.0 ⋯
b_a 10.0690 0.9636 0.0289 1120.0647 496.0030 1.0 ⋯
b_e 0.4998 0.2759 0.0082 1071.8186 549.0207 1.0 ⋯
b_i 1.5318 0.7030 0.0248 739.9494 359.8334 1.0 ⋯
b_ωy -0.0143 0.7142 0.0356 455.8501 720.8847 0.9 ⋯
b_ωx -0.0230 0.7112 0.0410 342.8093 840.4579 1.0 ⋯
b_Ωy 0.0533 0.7157 0.0337 492.5810 929.5566 0.9 ⋯
b_Ωx 0.0405 0.7106 0.0333 495.6578 732.6694 1.0 ⋯
b_θy -0.0234 0.7169 0.0371 417.2007 841.9656 1.0 ⋯
b_θx 0.0010 0.7004 0.0355 434.2525 783.5202 1.0 ⋯
b_ω -0.0245 1.8307 0.0934 453.4350 811.6041 0.9 ⋯
b_Ω 0.1040 1.7525 0.0762 582.9501 655.3153 0.9 ⋯
b_θ -0.0370 1.8457 0.0853 563.8540 837.9178 1.0 ⋯
b_tp 45123.4441 4220.0653 153.5460 814.9355 843.5914 0.9 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.9960 1.1316 1.1985 1.2659 1.3955
plx 49.9630 49.9854 49.9992 50.0135 50.0389
b_a 8.2241 9.3851 10.0664 10.7009 11.9726
b_e 0.0286 0.2677 0.5011 0.7272 0.9634
b_i 0.2621 0.9946 1.5035 2.0776 2.8005
b_ωy -1.0700 -0.7035 -0.0298 0.6846 1.0647
b_ωx -1.0531 -0.6970 -0.0636 0.6749 1.0568
b_Ωy -1.0728 -0.6373 0.1056 0.7374 1.0654
b_Ωx -1.0468 -0.6679 0.0865 0.7353 1.0591
b_θy -1.0752 -0.7226 -0.0088 0.6686 1.0599
b_θx -1.0332 -0.6846 -0.0295 0.6963 1.0435
b_ω -2.9839 -1.5834 -0.1366 1.6166 3.0291
b_Ω -2.9482 -1.3196 0.1584 1.5980 2.9908
b_θ -3.0146 -1.5971 -0.0466 1.5489 3.0011
b_tp 37926.2456 41249.8087 45349.0263 49438.5626 50403.3448
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.069043774384285
std(dat) = 0.9636174036592088Observable Based Priors
Octofitter implements observable-based priors from O'Neil 2019 for relative astrometry. You can fit a model to astrometry using observable-based priors using the following recipe:
using Octofitter, Distributions
astrom_like = PlanetRelAstromLikelihood(
(;epoch=mjd("2020-12-20"), ra=400.0, σ_ra=5.0, dec=400.0, σ_dec=5.0)
)
@planet b Visual{KepOrbit} begin
# For using with ObsPriors:
P ~ Uniform(0.001, 1000)
a = cbrt(system.M * b.P^2)
e ~ Uniform(0.0, 1.0)
i ~ Sine()
ω ~ UniformCircular()
Ω ~ UniformCircular()
mass ~ LogUniform(0.01, 100)
τ ~ UniformCircular(1.0)
tp = b.τ*b.P*365.25 + 58849 # reference epoch for τ. Choose an MJD date near your data.
end astrom_like ObsPriorAstromONeil2019(astrom_like);
@system System1 begin
plx ~ Normal(21.219, 0.060)
M ~ truncated(Normal(1.1, 0.2),lower=0.1)
end b