Likelihood package

Submodules

simple_pe.likelihood.likelihood module

simple_pe.likelihood.likelihood.like(a_hat, f_plus, f_cross, d_max=1000.0)

Return the likelihood marginalized over all 4 f-stat parameteras. Use uniform (1/2pi) prior on phi, psi, uniform (1/2)on cosi, uniform in volume prior over d out to d_max.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d_max – maximum distance for marginalization

simple_pe.likelihood.likelihood.like_approx(a_hat, f_plus, f_cross, d_max=1000.0, correction=False)

Calculate the approximate likelihood summed over left, right and coherent.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d_max – maximum distance for marginalization

• correction – scale the approx likelihood to improve fit

simple_pe.likelihood.likelihood.like_cosi(a_hat, f_plus, f_cross, x, d_max=1000.0)

Return the likelihood marginalized over d, phi and psi. Use uniform (1/2pi) prior on phi, psi, uniform in volume prior over d out to d_max.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• d_max – maximum distance for marginalization

simple_pe.likelihood.likelihood.like_cosi_psi(a_hat, f_plus, f_cross, x, psi, d_max=1000.0)

Return the likelihood marginalized over d and phi, using flat (1/2pi prior on phi; uniform volume on d up to d_max.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

• d_max – maximum distance for marginalization

simple_pe.likelihood.likelihood.like_d_cosi(a_hat, f_plus, f_cross, d, x)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood.like_d_cosi_psi(a_hat, f_plus, f_cross, d, x, psi, marg=True)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood.like_equal(a_hat, f, d_max=1000)

calculate the likelihood for network equally sensitive to plus and cross. Marginalized over the two phases, inclination and distance. We use uniform priors, specifically 1/2pi for psi,phi, 1/2 for cosi, 3 d^2dd/d_max^3 for distance d_max.

simple_pe.likelihood.likelihood.like_equal_cosi(a_hat, f, x, d_max=1000.0, make_plot=False)

For a network equally sensitive to plus and cross, calculate the likelihood marginalized over the distance and 2 phases. This a function of cosi. We use uniform priors, specifically 1/2pi for psi,phi, 1/2 for cosi, 3 d^2dd/d_max^3 for dist.

Parameters:

• a_hat – the f-stat parameters of the signal

• f – the detector response (f = F+ = Fx)

• x – cos(inclination) of template

• d_max – maximum distance for marginalization

• make_plot – plot the likelihood vs distance

simple_pe.likelihood.likelihood.like_equal_d_cosi(a_hat, f, d, cosi)

For a network equally sensitive to plus and cross, calculate the likelihood marginalized over the 2 phases. This a function of d and cosi. Note: we use uniform prior on phi, psi (=1/2pi)

Parameters:

• a_hat – the f-stat parameters of the signal

• f – the detector response (f = F+ = Fx)

• d – distance of template

• cosi – cos(inclination) of template

simple_pe.likelihood.likelihood.like_from_snr(zh, zl, sigh, sigl, dt, dt_ring, f_plus_cross)

Calculate the likelihood from observed SNRs for a 2 detector network

Parameters

zh: np.complex

The complex matched filter SNR for detector 1

zl: np.complex

The complex matched filter SNR for detector 2

sigh: float

sigma for detector 1

sigl: float

sigma for detector 2

dt: float

observed difference in merger time between the two detectors

dt_ring: np.ndarray

array containing the difference in merger time for ring in the sky

f_plus_cross: np.ndarray

array containing the detector sensitivity along the ring in the sky

simple_pe.likelihood.likelihood.like_parts_d_cosi_psi(a_hat, f_plus, f_cross, x, psi)

calculate the two dimensional likelihood, marginalized over phi log-likelihood can be written as: 1/2(ahat^2 - 2*d0/d * f(x, psi) * cos(2phi - phi0) + (d0/d)^2 g(x,psi)) return: ahat2, f, g

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

simple_pe.likelihood.likelihood.loglike_approx(a_hat, f_plus, f_cross, d_max=1000.0, method='coh', correction=False)

Calculate the approximate likelihood. This works for three cases: left and right circularly polarized and the standard coherent analysis. This reproduces the equations (A.18) and (A.29) from “Localization of transient gravitational wave sources: beyond triangulation”, Class. Quantum Grav. 35 (2018) 105002

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d_max – maximum distance for marginalization

• method – approximation for calculating likelihood, one of “coh”, “left”, “right”

• correction – scale the approx likelihood to improve fit

simple_pe.likelihood.likelihood_hm module

simple_pe.likelihood.likelihood_hm.l4422_d_cosi(a_hat, g_hat, f_plus, f_cross, d, x)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.l4422_d_cosi_psi(a_hat, g_hat, f_plus, f_cross, d, x, psi, marg=True)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.l4422_parts_d_cosi_psi(a_hat, g_hat, f_plus, f_cross, d, x, psi, phi)

calculate the two dimensional likelihood, marginalized over phi log-likelihood can be written as: 1/2(ahat^2 - 2*d0/d * f(x, psi) * cos(2phi - phi0) + (d0/d)^2 g(x,psi))

Returns:

ahat2, f, g

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

simple_pe.likelihood.likelihood_hm.l_d_c_phi(a_hat, f_plus, f_cross, d, x, phi, numerical=False)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.l_d_c_psi(a_hat, f_plus, f_cross, d, x, psi, numerical=False)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.lhm_d_cosi(a_hat, f_plus, f_cross, d, x, g_hat=None, j_hat=None, k_hat=None, cross_term_alphas={})

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.lhm_d_cosi_phi(a_hat, f_plus, f_cross, d, x, phi, g_hat=None, j_hat=None, k_hat=None, cross_term_alphas={})

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.lhm_d_cosi_psi(a_hat, f_plus, f_cross, d, x, psi, g_hat=None, j_hat=None, k_hat=None, cross_term_alphas={})

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.lhm_parts_d_cosi_psi(a_hat, f_plus, f_cross, d, x, psi, phi, g_hat=None, j_hat=None, k_hat=None, cross_term_alphas={})

calculate the two dimensional likelihood, marginalized over phi log-likelihood can be written as: 1/2(ahat^2 - 2*d0/d * f(x, psi) * cos(2phi - phi0) + (d0/d)^2 g(x,psi))

Returns:

ahat2, f, g

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

• g_hat – the 44 mode G Params

• j_hat – the 33 mode J Params

• k_hat – the 21 mode K Params

• cross_term_alphas – dictionary containing a tuple of the additional pair of

alpha values required for each mode (one each at 0 and pi/2. phase offset).

simple_pe.likelihood.likelihood_hm.like_22_d_cosi_psi_phi(a_hat, f_plus, f_cross, d, x, psi, phi)

Return the un-marginalized likelihood for a 2,2 waveform model.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• phi – coalescence phase

simple_pe.likelihood.likelihood_hm.like_cosi_alpha_44(a_hat, g_hat, f_plus, f_cross, x, alpha_44_prime, d_max=1000)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_cosi_alpha_44_phi(a_hat, g_hat, f_plus, f_cross, x, phi, alpha_44_prime, d_max)

Return the likelihood marginalized over d and psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_cosi_alpha_44_psi_phi(a_hat, g_hat, f_plus, f_cross, x, psi, phi, alpha_44_prime, d_max, k=256)

Likelihood marginalized over distance using 3/d_max**3 d**2 dd prior. Currently no prior on alpha_44 (k exponential) is implemented.

simple_pe.likelihood.likelihood_hm.like_d(a_hat, f_plus, f_cross, d)

Return the likelihood marginalized over phi, psi, with a uniform (1/2pi) prior on both, and cosi with uniform 1/2 prior.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

simple_pe.likelihood.likelihood_hm.like_d_alpha_33(a_hat, j_hat, f_plus, f_cross, d, alpha_33_prime)

Return the likelihood marginalized over phi, psi, with a uniform (1/2pi) prior on both, and cosi with uniform 1/2 prior.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_d_alpha_33_alpha_44(a_hat, j_hat, g_hat, f_plus, f_cross, d, alpha_33_prime, alpha_44_prime)

Return the likelihood marginalized over phi, psi, with a uniform (1/2pi) prior on both, and cosi with uniform 1/2 prior.

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

simple_pe.likelihood.likelihood_hm.like_d_cosi(a_hat, f_plus, f_cross, d, x)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_d_cosi_2(a_hat, f_plus, f_cross, d, x)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33(a_hat, j_hat, g_hat, f_plus, f_cross, d, x, alpha_33_prime, alpha_44_prime)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_alpha_44(a_hat, j_hat, g_hat, f_plus, f_cross, d, x, alpha_33_prime, alpha_44_prime)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_alpha_44_phi(a_hat, j_hat, g_hat, f_plus, f_cross, d, x, phi, alpha_33_prime, alpha_44_prime)

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_alpha_44_psi_phi(a_hat, j_hat, g_hat, f_plus, f_cross, d, x, psi, phi, alpha_33_prime, alpha_44_prime)

unmarginalized likelihood over five dimensions for signal model containing 22, 33 and 44 modes.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_phi(a_hat, j_hat, f_plus, f_cross, d, x, phi, alpha_33_prime)

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_phi_with_noise(a_hat, j_hat, f_plus, f_cross, d, x, phi, alpha_33_prime)

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_psi_phi(a_hat, j_hat, f_plus, f_cross, d, x, psi, phi, alpha_33_prime)

unmarginalized likelihood over five dimensions for signal model containing 22 and 33 modes.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_psi_phi_with_noise(a_hat, j_hat, f_plus, f_cross, d, x, psi, phi, alpha_33_prime)

unmarginalized likelihood over five dimensions for signal model containing 22 and 33 modes. Containing gaussian noise in the matched filter SNR term that appears in the likelihood.

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_33_with_noise(a_hat, j_hat, f_plus, f_cross, d, x, alpha_33_prime)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_d_cosi_alpha_44_psi_phi(a_hat, g_hat, f_plus, f_cross, d, x, psi, phi, alpha_44_prime, k=256)

unmarginalized likelihood over five dimensions for signal model containing 22 and 44 modes.

simple_pe.likelihood.likelihood_hm.like_d_cosi_phi(a_hat, f_plus, f_cross, d, x, phi, marg=True)

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_d_cosi_psi(a_hat, f_plus, f_cross, d, x, psi, marg=True)

Return the likelihood marginalized over phi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_marg_over_alpha_44(a_hat, g_hat, f_plus, f_cross, d, x, alpha_44_hat, marg=True)

Return the likelihood marginalized over phi and psi, with a uniform (1/2pi) prior on both

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

simple_pe.likelihood.likelihood_hm.like_marg_over_alpha_44_phi(a_hat, g_hat, f_plus, f_cross, d, x, phi, alpha_44_hat, marg)

Return the likelihood marginalized over psi, using flat (1/2pi) prior

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• d – distance

• x – cos(inclination)

• psi – polarization

• marg – do or don’t do the marginalization

Returns:

the marginalization factor. I don’t think it includes the exp(rho^2/2) term.

simple_pe.likelihood.likelihood_hm.like_marg_over_alpha_44_psi_phi(a_hat, g_hat, f_plus, f_cross, d, x, psi, phi, alpha_44_hat, marg)

to be used by below function

int_0^inf L22*k*exp(-a alpha_44^2 + b alpha_44) dalpha44 = L22 * k * sqrt(pi) * exp(b**2/(4*a)) * ( erf(b/(2*sqrt(a))) + 1 ) / (2*sqrt(a))

—the last line can be re-written using the complementary error function, cerf=1-erf, and we can use the exponentially scaled version to prevent rounding errors. AND must add the data ghat2 factor to normalize.

simple_pe.likelihood.likelihood_hm.like_parts_d_cosi_psi(a_hat, f_plus, f_cross, x, psi)

calculate the two dimensional likelihood, marginalized over phi log-likelihood can be written as: 1/2(ahat^2 - 2*d0/d * f(x, psi) * cos(2phi - phi0) + (d0/d)^2 g(x,psi))

Returns:

ahat2, f, g

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

simple_pe.likelihood.likelihood_hm.like_parts_d_cosi_psi_phi(a_hat, f_plus, f_cross, d, x, psi, phi, numerical=False, phi_first=False)

calculate the two dimensional likelihood, marginalized over phi log-likelihood can be written as: 1/2(ahat^2 - 2*d0/d * f(x, psi) * cos(2phi - phi0) + (d0/d)^2 g(x,psi))

Returns:

ahat2, f, g

Parameters:

• a_hat – the F-stat A parameters

• f_plus – F_plus sensitivity

• f_cross – F_cross sensitivity

• x – cos(inclination)

• psi – polarization

simple_pe.likelihood.likelihood_hm.sqri21(iota)

simple_pe.likelihood.likelihood_hm.sqri2m1(iota)

simple_pe.likelihood.likelihood_hm.sqri32(iota)

simple_pe.likelihood.likelihood_hm.sqri33(iota)

simple_pe.likelihood.likelihood_hm.sqri3m2(iota)

simple_pe.likelihood.likelihood_hm.sqri3m3(iota)

simple_pe.likelihood.likelihood_hm.sqri43(iota)

simple_pe.likelihood.likelihood_hm.sqri4m3(iota)

simple_pe.likelihood.likelihood_hm.t(iota)

simple_pe.likelihood.likelihood_hm.temp_signal_likelihood(a_temp, a_hat, f_resp_2, alpha_hm_prime=1)