Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hea-www.harvard.edu/AstroStat/astro193/VLK_slides_mar04.pdf Äàòà èçìåíåíèÿ: Fri Mar 6 04:02:25 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 11:58:33 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

Astro 193 : 2015 Mar 4
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Follow-up
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Extra class/doodle poll MCMC Assumptions

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Transformation of variables -- the Jacobian Aperture Photometry Homework 7 Bias and Measurement Errors


MCMC Assumptions

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A stationary or invariant distribution must exist The chain must be homogeneous, and reversible The system must be ergodic: any point in the parameter space must be reachable from any other point
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Ergodicity implies the chain must converge


Transformations
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When variables ={, i=1..n} are transformed to ={, i=1..n}, the volumes transform as
i=1..n

d = det[/] =1..n d

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e.g., Cartesian coordinates (x,y) transform to polar coordinates (r,) with Jacobian r; dxdy = rdrd e.g., (S,H) (R=S/H, H) dS dH = H dR dH Quasi Homework: How would you deal with HR = (H - S)/(H + S) ?

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Bayesian Aperture Photometry (Gaussian)

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Note: the following slides differ in content and order from what was shown in class. They have been edited to clear up some notational errors, and explicitly include the effect of an informational prior on the posterior density of the source intensity.


Product of two Gaussians
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g = N(µ,2) N(µ,2) exp[-(x-µ)²/2² - (x-µ)²/2²]

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µ = (µ²+µ²) / (²+²) ; ² = ²² / (²+²) for , µ µ and

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g = (1/2) exp[-(x-µ)²/2²] (1/2(²+²)) exp[(µ-µ)²/2(²+²)]


Bayesian Aperture Photometry (Gaussian)
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Measurements: Background in an area AB = rAS: fB ±
B M

(Source+Background) in an area AS: fM ±
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Priors: p(b) = N(b;f0B,0B²) p(s) = N(s;f0S,0S²)

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Likelihoods: f(fB,B|b,r) exp[-(rb-fB)²/2B²] exp[-(b-(fB/r))²/2(B/r)²] exp[-(b-fB)²/2B²] f(fM,M|s,b) exp(-((s+b)-fM)²/2M²]


Bayesian Aperture Photometry (Gaussian)
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p(b|fB,B,r) p(b) p(fB,B|b,r) exp[-(b-gB)²/2g²] with gB = (f0BB² + fB0B²)/(B²+B²), g =B²0B²/(B²+B²)
2

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p(sb|fB,B,fM,M,r) p(sb) p(fB,B,fM,M|s,b.r) p(b) p(s|b) p(fBB|s,b,r) p(fMM|fB,B,s,b,r) p(b) p(fBB|b,r) p(s) p(fMM|s,b,r) exp[-(b-gB)²/2g²] exp[-(s-f0S)²/20S²] exp[-(s+b-fM)²/2M²] exp[-(b-gB)²/2g² - (b-(fM-s))²/2M²] exp[-(s-f0S)²/20S²] exp[-(b-hB)²/2wB²] exp[-(gB-(fM-s))²/2(g²+M²)] exp[-(s-f0S)²/20S²] with hB=(gBM²+(fM-s)g²)/2(g²+M²) and wB=M²g²/(g²+M²) exp[-(b-hB)²/2wB²] exp[-(s-(fM-gB))²/2(g²+M²)] exp[-(s-f0S)²/20S²] exp[-(b-hB)²/2wB²] exp[-(s-hS)²/2wS²] exp[-(fM-gB-f0S)²/2(g²+M²+0S²)] with hS=((fM-gB)0S²+f0S(g²+M²)/(0S²+g²+M²) and wS=0S²(g²+M²)/(0S²+g²+M²) exp[-(b-hB)²/2wB²] exp[-(s-hS)²/2wS²]

· Marginalizing over b, p(s|fB,B,fM,M,r) exp[(s-hS)²/2wS²]


Bayesian Aperture Photometry (Gaussian)
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When the prior distributions on s and b are very broad and non-informative gB fB, g
B

hS fM-fB, wS² M²+B² hB (fBM² + (fM-s)B²)/(M²+B²) and wB² S²B²/(M²+B²) i.e., the estimate for the mean background is the error-weighted mean of the background estimates from the background aperture and the source+background aperture
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and the marginalized posterior density for s, p(s|fB,B,fM,M,r) exp[(s-(fM-fB))²/2(B²+M²)] implies that the mean intensity of the source is simply the difference between the measured intensity in the source+background region and the area-scaled measured intensity of the background, and the error on this background-subtracted source intensity is the square-added measurement errors. This is exactly the result from frequentist or likelihood analysis (see Homework 2).