# 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:

• Normal
• Uniform
• LogNormal
• LogUniform
• TrucatedNormal
• VonMises

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.

## Observable 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)
end b

model = Octofitter.LogDensityModel(System1, verbosity=4)
LogDensityModel for System System1 of dimension 12 with fields .ℓπcallback and .∇ℓπcallback

octofit(
model, 0.95;
iterations = 1000,
verbosity = 4,
max_depth = 10,
)
Chains MCMC chain (1000×30×1 Array{Float64, 3}):

Iterations        = 1:1:1000
Number of chains  = 1
Samples per chain = 1000
Wall duration     = 21.89 seconds
Compute duration  = 21.89 seconds
parameters        = plx, M, b_P, b_e, b_i, b_ωy, b_ωx, b_Ωy, b_Ωx, b_mass, b_τy, b_τx, b_a, b_ω, b_Ω, b_τ, 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      rhat  ⋯
Symbol      Float64   Float64   Float64    Float64    Float64   Float64  ⋯

plx      21.1664    0.0000       NaN        NaN        NaN       NaN  ⋯
M       1.1068    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_P     405.1178    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_e       0.5281    0.0000       NaN        NaN        NaN       NaN  ⋯
b_i       1.3743    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_ωy       1.2599    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_ωx      -0.1120    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_Ωy       0.6406    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_Ωx       0.6492    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_mass       6.6210    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_τy       0.9663    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_τx       0.0145    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_a      56.6347    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_ω      -0.0886    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_Ω       0.7921    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_τ       0.0024    0.0000    0.0000        NaN        NaN       NaN  ⋯
b_tp   59203.0000    0.0000       NaN        NaN        NaN       NaN  ⋯
1 column omitted

Quantiles
parameters         2.5%        25.0%        50.0%        75.0%        97.5%
Symbol      Float64      Float64      Float64      Float64      Float64

plx      21.1664      21.1664      21.1664      21.1664      21.1664
M       1.1068       1.1068       1.1068       1.1068       1.1068
b_P     405.1178     405.1178     405.1178     405.1178     405.1178
b_e       0.5281       0.5281       0.5281       0.5281       0.5281
b_i       1.3743       1.3743       1.3743       1.3743       1.3743
b_ωy       1.2599       1.2599       1.2599       1.2599       1.2599
b_ωx      -0.1120      -0.1120      -0.1120      -0.1120      -0.1120
b_Ωy       0.6406       0.6406       0.6406       0.6406       0.6406
b_Ωx       0.6492       0.6492       0.6492       0.6492       0.6492
b_mass       6.6210       6.6210       6.6210       6.6210       6.6210
b_τy       0.9663       0.9663       0.9663       0.9663       0.9663
b_τx       0.0145       0.0145       0.0145       0.0145       0.0145
b_a      56.6347      56.6347      56.6347      56.6347      56.6347
b_ω      -0.0886      -0.0886      -0.0886      -0.0886      -0.0886
b_Ω       0.7921       0.7921       0.7921       0.7921       0.7921
b_τ       0.0024       0.0024       0.0024       0.0024       0.0024
b_tp   59203.0000   59203.0000   59203.0000   59203.0000   59203.0000