Issue 
A&A
Volume 574, February 2015



Article Number  A136  
Number of page(s)  20  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201424451  
Published online  10 February 2015 
Online material
Appendix A: Most probable in general case
The MP2 estimators, and , have to satisfy Eqs. (8) and (9) simultaneously. These relations can be solved using the fully developed expression of f_{2D}, including the terms of the inverse matrix : (A.1)leading to (A.2)with (A.3)This analytical solution only depends on the input measurements (p, ψ) and the covariance matrix Σ_{p}. Because the polarization fraction must be positive, there is a lower limit of the S/N so that . In that case, is not constrained anymore and can be chosen to be any possible value. We set it equal to
the measurement ψ. Moreover, this expression can be simplified when ρ = 0, which implies that v_{12} = 0, leading to (A.4)In the canonical case (v_{12} = 0, ), we recover the expression derived by Quinn (2012): (A.5)
Appendix B: Bayesian posterior PDF
We illustrate the shape of the posterior PDF in Fig. B.1, where B_{2D}(p_{0},ψ_{0}  p,ψ,Σ_{p}) is shown at four levels of the S/N and five couples of (ε, ρ). It is interesting to notice that the posterior PDF allows the polarization fraction to be zero at low S/N, when these values were rejected by the PDF (see Appendix B of PMA I). Moreover, the posterior PDF peaks at the location of the measurements used to compute it. As largely emphasized in PMA I, we also recall that once the effective ellipticity of the covariance matrix departs from the canonical simplification, the PDFs are sensitive to the initial true polarization angle ψ_{0}.
Fig. B.1
Posterior probability density functions B_{2D}(p_{0},ψ_{0}  p,ψ,Σ_{p}) computed for the most probable measurements (p, ψ) of the f_{2D} distribution (crosses), which were obtained for a given set of true polarization parameters ψ_{0} = 0° and p_{0} = 0.10 (dashed lines) and various configurations of the covariance matrix, at four levels of S/N p_{0}/σ_{p,G} = 0.1,0.5,1, and 5 (top to bottom). The scales of the p_{0} and ψ_{0} axes may vary from one row to the next in order to focus on the interesting part of the PDF. The black contours provide the 90, 70, 50, 20, 10, 5, 1, and 0.1% levels. 

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Appendix C: Mean Bayesian posterior analytical expression
In the canonical case, the MB estimator of the polarization fraction p takes a simple analytical expression. The Bayesian posterior on p is given in this case by (C.1)where κ is the prior chosen equal to one over the definition range ([0,1]), and R denotes the Rice (1945) function which is defined by (C.2)where ℐ_{0}(x) is the zerothorder modified Bessel function of the first kind (Gradshteyn & Ryzhik 2007), and σ_{p} = σ_{Q}/I_{0} = σ_{U}/I_{0} is the characteristic noise level of the polarization fraction.
The MB estimator and the posterior variance take the following forms (C.3)and (C.4)If we assume in a first approximation that the integral of p_{0} over [ 0,1 ] can be taken over [ 0, + ∞) (which is fine at high S/N), and we use the formula of Prudnikov et al. (1986), (C.5)where Γ is the Gamma function, the confluent hypergeometric function of the first kind, and a, b, and c all positive reals, we can derive (C.6)\pagebreak
and (C.7)and finally (C.8)We finally obtain the simple expression of the MB estimator and the associated Bayesian variance: (C.9)and (C.10)As shown in Fig. C.1, this analytical approximation gives less than 0.15% of relative error at low S/N compared to the exact estimate and less than 0.05% for the associated uncertainty. This small departure quickly tends to 0 for a S/N> 4. Thus these expressions may be used to speed up the computing time when the canonical simplification may be assumed.
Fig. C.1
Accuracy of the approximate analytical expression of the Bayesian estimates of the polarization fraction (solid line) and its associated uncertainty (dashed line), as a function of the S/N of the measurement p/σ_{p}, where σ_{p} = σ_{Q}/I_{0} = σ_{U}/I_{0}. 

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Fig. C.2
Accuracy of the generalized approximate analytical expression of the Bayesian estimates (top) and (bottom), taking the full covariance matrix components into account, in the low (light grey) and tiny (dark grey) regimes. 

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We explore in Fig. C.2 to the extent at which the canonical simplification may be done in the presence of an effective ellipticity of the covariance matrix. In this more general case, we suggest changing σ_{p} into σ_{p,G} in the Eqs. (C.9) and (C.10). The relative error between the approximate estimate and the exact Bayesian estimate has been explored in two regimes of the covariance matrix, the low (1 <ε_{eff}< 1.1) and tiny (1 <ε_{eff}< 1.01) regimes. Three domains are observed in the top panel of Fig. C.2 dealing with the accuracy of the estimate: i) at low S/N (<1), the bias on p is so large that the presence of an effective ellipticity does not significantly affect the estimate in comparison; ii) for an intermediate range of the S/N (1 <S/N< 4), the effective ellipticity of the Σ_{p} significantly affects the Bayesian estimate so that the departure of the analytical approximation from the exact estimate becomes important; iii) at high S/N (>4), the noise is so low that the Bayesian estimate is not sensitive to the asymmetry of the covariance matrix anymore. Consequently, the approximate analytical expression provides very good estimates of for S/N< 1 and S/N> 4, and 5% to 0.5% of relative error for intermediate 1 <S/N< 4 in the low and tiny regimes of the covariance matrix, respectively. In the extreme regime of the covariance matrix, the relative error increases up to 20%.
Concerning the accuracy of the Bayesian approximate estimate of the polarization fraction uncertainty (bottom panel), the agreement is better than 0.1% for S/N< 1, and about 8% S/N> 1 in the low regime, and 1% in the tiny regime. Because the uncertainty becomes small compared to the polarization fraction at high S/N, up to 8% of error in is still acceptable for this approximation.
© ESO, 2015
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