Priors

Default Priors

Parameter	Support / Integration Limits	Prior Type	Analytical Form
m1	[1,100]	Uniform	\(p(m_1)\propto 1\)
m2	[1,100]	Uniform	\(p(m_2)\propto 1\)
M	[1,200]	Uniform	\(p(M)\propto 1\)
Mc	[1,100]	Uniform	\(p(\mathcal{M}_c)\propto 1\)
ln_Mc	\([ln\ 1,ln\ 100]\)	Log-chirp-mass prior	\(p(\ln\mathcal{M}_c)\propto e^{\ln\mathcal{M}_c}\)
q	\([10^{-2},1-\epsilon]\)	Mass-ratio prior	\(p(q)\propto q^2\)
eta	derived from q	Derived prior	Derived from the prior on q
inv_eta	derived from eta	Derived prior	Derived from the prior on eta
ln_eta	derived from eta	Derived prior	Derived from the prior on eta
deltaM	derived from eta	Derived prior	Derived from the prior on eta
RA	\([0,360^o]\)	Uniform	\(p(\alpha)\propto 1\)
Dec	\([-90^o,90^o]\)	Isotropic sky prior	\(p(\delta)\propto \cos\delta\)
iota	\([0,\pi]\)	Sine prior	\(p(\iota)\propto \sin\iota\)
cos_iota	[-1,1]	Uniform	\(p(\cos\iota)\propto 1\)
psi	\([0,\pi]\)	Uniform	\(p(\psi)\propto 1\)
t_coal	[-1,1]	Uniform	\(p(t_{\rm coal})\propto 1\)
phi_coal	\([-\pi,\pi]\)	Uniform	\(p(\phi_{\rm coal})\propto 1\)
sx1,sy1,sz1	[-0.98,0.98]	Uniform	\(p(s_i)\propto 1\)
sx2,sy2,sz2	[-0.98,0.98]	Uniform	\(p(s_i)\propto 1\)
chi1	[0,0.98]	Uniform	\(p(\chi_1)\propto 1\)
phi1	\([0,2\pi]\)	Uniform	\(p(\phi_1)\propto 1\)
theta1	\([0,\pi]\)	Spherical-angle prior	\(p(\theta_1)\propto \sin\theta_1\)
chi2	[0,0.98]	Uniform	\(p(\chi_2)\propto 1\)
phi2	\([0,2\pi]\)	Uniform	\(p(\phi_2)\propto 1\)
theta2	\([0,\pi]\)	Spherical-angle prior	\(p(\theta_2)\propto \sin\theta_2\)
chi_s	[-0.98,0.98]	Uniform	\(p(\chi_s)\propto 1\)
chi_a	[-0.98,0.98]	Uniform	\(p(\chi_a)\propto 1\)
dL	cosmology-dependent	Astrophysical distance prior	\(p(d_L)\propto \frac{dV_c}{dd_L}\frac{\mathrm{SFR}(z)}{1+z}\)
lnDL	derived from dL	Derived prior	Derived from the prior on dL
inv_dL	derived from dL	Derived prior	Derived from the prior on dL
inv_dL2	derived from dL	Derived prior	Derived from the prior on dL
inv_sqrtdL	derived from dL	Derived prior	Derived from the prior on dL
inv_lnDL	derived from dL	Derived prior	Derived from the prior on dL

Make your own prior

To redefine your priors define a dictionary with two arrays for each parameter, one for the support and another for the probability distribution. For instance, to redefine the priors on dL, iota, Mc and eta as shown in the figure below, we can run the code snippet below.

import numpy as np
import GWDALI as gw
import matplotlib.pyplot as plt

FreeParams = "dL,Mc,eta,iota".split(',')

Priors_default = gw.Priors(FreeParams,name=None,new_priors=None,plot=False)

dL   = np.linspace(.1,250,1000)
iota = np.linspace(0,np.pi,1000)
Mc   = np.linspace(1,100,1000)
eta  = np.linspace(0.0,0.25,1000)

l0 = np.pi/6

p_dL = np.exp(-.5*(dL-50)**2/50**2) ; p_dL/=sum(p_dL)
p_Mc = Mc**(-.5)+np.exp(-0.5*((Mc-50)/5)**2)/np.sqrt(2*np.pi*5**2) ; p_Mc/=sum(p_Mc)
p_eta = eta**2 ; p_eta/=sum(p_eta)
p_iota = np.sin(iota)*(iota<l0)+np.sin(iota)*(iota>np.pi-l0) ; p_iota/=sum(p_iota)

new_priors = {}
new_priors["dL"] = [dL,p_dL]
new_priors["iota"] = [iota,p_iota]
new_priors["Mc"] = [Mc,p_Mc]
new_priors["eta"] = [eta,p_eta]

Priors_modified = gw.Priors(FreeParams,new_priors=new_priors,plot=False)

for key in FreeParams:
        p1 = Priors_default[key]
        p2 = Priors_modified[key]

        x1 = np.linspace(p1.minimum,p1.maximum,1000)
        x2 = np.linspace(p2.minimum,p2.maximum,1000)
        y1 = p1.prob(x1) ; y1/=max(y1)
        y2 = p2.prob(x2) ; y2/=max(y2)

        plt.subplot(2,2,FreeParams.index(key)+1)
        plt.plot(x1,y1,'k-',label="Default Prior")
        plt.plot(x2,y2,'r-',label="New Prior")
        plt.xlabel(key)
        plt.legend()
        plt.grid(alpha=.3)

plt.tight_layout()
plt.show()