. Author manuscript; available in PMC: 2020 Aug 26.

Published in final edited form as: Astrophys J. 2018 Aug 7;863(1):15. doi: 10.3847/1538-4357/aace62

Table 5.

Priors and Posteriors of Bayesian Analysis

Parameter	Prior	Mean of Posterior	Description
Case 1: K power-law/M power-law model
γ₁	flat^a on [1.2, 3.5]	${2.21}_{- 0.11}^{+ 0.12}$	primary PSD slope
γ₂^b	flat^a on [1.2, 10.0]	${6.0}_{- 2.5}^{+ 2.8}$	secondary PSD slope
f_b [10⁻³ minutes⁻¹]	Flat^v on [1.0, 600.0]	${3.50}_{- 0.89}^{+ 0.98}$	primary correlation frequency
f_b,2 [minutes⁻]	flat^a on [0.001, 0.6]	${0.34}_{- 0.16}^{+ 0.18}$	secondary break frequency
		>0.120	(95% credible level)
F₀ [mJy]	Gaussian (p = −0.36, o = 0.05)	$- {0.37}_{- 0.05}^{+ 0.05}$	pole of the power-law flux-density PDF (K- and M-band)
β_k	Gaussian (p = 4.22, o = 0.6)	${4.53}_{- 0.33}^{+ 0.34}$	slope of the power-law flux-density PDF in K-band
β_m	Gaussian (p = 4.22, o = 0.6)	${4.45}_{- 0.56}^{+ 0.56}$	slope of the power-law flux-density PDF in M-band
s	flat on [0.01, 24.0]	${5.9}_{- 1.9}^{+ 2.5}$	M to K flux density ratio
σ_Keck [mJy]	Gaussian (p = 0.017, o = 0.008)	${0.014}_{- 0.006}^{+ 0.004}$	measurement noise of the Keck observations
σ_VLT [mJy]	Gaussian (p = 0.034, o = 0.008)	${0.031}_{- 0.003}^{+ 0.003}$	measurement noise of the VLT observations
σ_IRAC [mJy]	Gaussian (p = 0.65, o = 0.2)	${0.676}_{- 0.019}^{+ 0.020}$	measurement noise of the IRAC observations
Case 2: K power-law/M log-normal model
γ₁	flat^a on [1.2, 3.5]	${2.21}_{- 0.11}^{+ 0.12}$	primary PSD slope
γ₂^b	flat^a on [1.2, 10.0]	${6.0}_{- 2.6}^{+ 2.7}$	secondary PSD slope
f_b [10⁻³ minutes⁻¹]	flat^a on [1.0, 600.0]	${3.71}_{- 1.1}^{+ 1.2}$	primary correlation frequency
f_b,2 [minutes⁻¹]	flat^a on [0.001, 0.6]	${0.33}_{- 0.16}^{+ 0.17}$	secondary break frequency
		>0.112	(95% credible level)
F₀ [mJy]	Gaussian (μ = −0.37, o = 0.05)	$- {0.37}_{- 0.04}^{+ 0.05}$	pole of the power-law flux-density PDF in K-band
β_K	Gaussian (μ = 4.22, o = 0.6)	${4.59}_{- 0.32}^{+ 0.32}$	slope of the power-law flux-density PDF in K-band
μ_logn,M	flat on [−6.0, 6.0]	$- {0.3}_{- 1.3}^{+ 1.3}$	log-normal mean in M-band
σ_logn,M	flat on [0.001, 4.0]	${0.89}_{- 0.50}^{+ 0.54}$	log-normal standard deviation in M-band
σ_Keck [mJy]	Gaussian (μ = 0.017, o = 0.008)	${0.015}_{- 0.006}^{+ 0.004}$	measurement noise of the Keck observations
σ_VLT [mJy]	Gaussian (μ = 0.034, o = 0.008)	${0.031}_{- 0.003}^{+ 0.003}$	measurement noise of the VLT observations
σ_IRAC [mJy]	Gaussian (μ = 0.65, o = 0.2)	${0.678}_{- 0.019}^{+ 0.020}$	measurement noise of the IRAC observations
Case 3: K log-normal/M log-normal model + spectral information from synchronous data
γ₁	flat^a on [1.2, 3.5]	${2.10}_{- 0.09}^{+ 0.10}$	primary PSD slope
γ₂^b	flat^a on [1.2, 10.0]	${5.8}_{- 2.4}^{+ 2.8}$	secondary PSD slope
f_b [10⁻³ minutes⁻¹]	flat^a on [1.0, 600.0]	${4.11}_{- 0.65}^{+ 0.76}$	primary correlation frequency
f_b,2 [minutes⁻¹]	flat^a on [0.001, 0.6]	${0.31}_{- 0.15}^{+ 0.19}$	secondary break frequency
		>0.118	(95% credible level)
μ_logn,K	flat on [−8.3, 3.7]	$- {1.35}_{- 0.60}^{+ 0.62}$	log-normal mean in K-band
σ_logn,K	flat on [0.001, 4.0]	${0.56}_{- 0.21}^{+ 0.24}$	log-normal standard deviation in K-band
μ_logn,M	flat on [−6.0, 6.0]	${1.01}_{- 0.44}^{+ 0.47}$	log-normal mean in M-band
σ_logn,M	flat on [0.001, 4.0]	${0.39}_{- 0.13}^{+ 0.15}$	log-normal standard deviation in M-band
OKeck [mJy]	Gaussian (μ = 0.017, o = 0.008)	${0.013}_{- 0.006}^{+ 0.005}$	measurement noise of the Keck observations
σ_VLT [mJy]	Gaussian (μ = 0.034, o = 0.008)	${0.030}_{- 0.003}^{+ 0.002}$	measurement noise of the VLT observations
σ_IRAC [mJy]	Gaussian (μ = 0.65, o = 0.2)	${0.677}_{- 0.013}^{+ 0.013}$	measurement noise of the IRAC observations

Notes.

The joint prior distributions are flat under the conditions f_b,2 > f_b and γ₂ > γ₁, respectively; see Appendix B.3.

Unconstrained by the data; posterior is a minor alteration of the prior.