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 = Planet(
name="b",
basis=Visual{KepOrbit},
observations=[],
variables=@variables begin
a ~ kde # Sample from the KDE here
e ~ Uniform(0.0, 0.99)
i ~ Sine()
M = system.M
ω ~ UniformCircular()
Ω ~ UniformCircular()
θ ~ UniformCircular()
tp = θ_at_epoch_to_tperi(θ, 50000; M, e, a, i, ω, Ω)
end
)
sys = System(
name="Tutoria",
companions=[planet_b],
observations=[],
variables=@variables begin
M ~ truncated(Normal(1.2, 0.1), lower=0.1)
plx ~ truncated(Normal(50.0, 0.02), lower=0.1)
end
)
model = Octofitter.LogDensityModel(sys)
chain = octofit(model)Chains MCMC chain (1000×29×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 0.75 seconds
Compute duration = 0.75 seconds
parameters = M, plx, b_a, b_e, b_i, b_ωx, b_ωy, b_Ωx, b_Ωy, b_θx, b_θy, b_ω, b_Ω, b_θ, b_M, 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 1.2003 0.0978 0.0033 872.5233 630.2477 1.0 ⋯
plx 49.9984 0.0196 0.0006 1183.6881 639.6990 0.9 ⋯
b_a 10.0334 1.0885 0.0362 901.2056 475.1670 0.9 ⋯
b_e 0.5166 0.2818 0.0071 1508.7590 680.6235 1.0 ⋯
b_i 1.5938 0.6773 0.0213 985.9787 675.8711 0.9 ⋯
b_ωx -0.0250 0.7280 0.0426 335.1399 809.2381 1.0 ⋯
b_ωy -0.0428 0.7059 0.0411 338.6026 843.2654 1.0 ⋯
b_Ωx -0.0464 0.7187 0.0381 409.9114 814.0048 1.0 ⋯
b_Ωy -0.0194 0.6977 0.0388 349.2600 863.9743 0.9 ⋯
b_θx 0.0462 0.7368 0.0357 419.0764 713.7857 1.0 ⋯
b_θy 0.0042 0.6878 0.0332 438.5318 820.8215 0.9 ⋯
b_ω -0.0550 1.8442 0.0822 666.0001 866.8385 1.0 ⋯
b_Ω -0.1149 1.8738 0.0882 553.0694 717.5318 0.9 ⋯
b_θ 0.0088 1.7861 0.0732 674.9194 793.0973 1.0 ⋯
b_M 1.2003 0.0978 0.0033 872.5233 630.2477 1.0 ⋯
b_tp 44597.3326 4288.6715 152.4061 841.2841 748.7792 0.9 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 1.0046 1.1370 1.2019 1.2610 1.3904
plx 49.9597 49.9855 49.9983 50.0113 50.0368
b_a 7.8352 9.2849 10.1058 10.7510 12.1964
b_e 0.0366 0.2764 0.5213 0.7543 0.9651
b_i 0.3179 1.0767 1.6357 2.1293 2.7748
b_ωx -1.0609 -0.7418 -0.0517 0.7119 1.0818
b_ωy -1.0702 -0.7208 -0.0779 0.6123 1.0636
b_Ωx -1.0588 -0.7702 -0.0958 0.6726 1.0415
b_Ωy -1.0451 -0.6974 -0.0418 0.6333 1.0471
b_θx -1.0571 -0.7113 0.1244 0.7660 1.0971
b_θy -1.0614 -0.6473 0.0200 0.6368 1.0479
b_ω -2.9595 -1.6279 -0.1555 1.5975 3.0287
b_Ω -3.0489 -1.8017 -0.1320 1.4826 2.9872
b_θ -2.9804 -1.5227 0.0397 1.4038 2.9948
b_M 1.0046 1.1370 1.2019 1.2610 1.3904
b_tp 37522.5607 40729.8556 44448.6021 49125.0180 49978.9296
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.03341008938257
std(dat) = 1.0884597240705292Observable 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_dat = Table(
epoch=[mjd("2020-12-20")],
ra=[400.0],
σ_ra=[5.0],
dec=[400.0],
σ_dec=[5.0]
)
astrom_obs = PlanetRelAstromObs(astrom_dat, name="rel astrom. 1")
planet_b = Planet(
name="b",
basis=Visual{KepOrbit},
observations=[ObsPriorAstromONeil2019(astrom_obs)],
variables=@variables begin
# For using with ObsPriors:
P ~ Uniform(0.001, 1000)
M = system.M
a = cbrt(M * P^2)
e ~ Uniform(0.0, 1.0)
i ~ Sine()
ω ~ UniformCircular()
Ω ~ UniformCircular()
mass ~ LogUniform(0.01, 100)
τ ~ UniformCircular(1.0)
tp = τ*P*365.25 + 58849 # reference epoch for τ. Choose an MJD date near your data.
end
)
sys = System(
name="System1",
companions=[planet_b],
observations=[],
variables=@variables begin
plx ~ Normal(21.219, 0.060)
M ~ truncated(Normal(1.1, 0.2),lower=0.1)
end
)