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Course on Inverse Problems Albert Tarantola Third Lesson: - - PowerPoint PPT Presentation
Course on Inverse Problems Albert Tarantola Third Lesson: - - PowerPoint PPT Presentation
Princeton University Department of Geosciences Course on Inverse Problems Albert Tarantola Third Lesson: Probability (Elementary Notions) Let u and v be two Cartesian parameters (then, volumetric probabilities and probability densities are
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0.02 0.04 0.06 0.08 0.1 0.12
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5 10 0.025 0.05 0.075 0.1 0.125 0.15
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5 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14
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5 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14
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5 10 0.05 0.1 0.15 0.2 0.25
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5 10 0.05 0.1 0.15 0.2 0.25
joint conditional conditional conditional conditional marginal marginal
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Let u and v be two Cartesian parameters (then, volumetric probabilities and probability densities are identical). Given a probability density f (u, v) , one defines the two marginal probability densities fu(u) =
+∞
−∞ dv f (u, v)
fv(v) =
+∞
−∞ du f (u, v)
and the two conditional probability densities fu|v(u|v = v0) = f (u, v0)
+∞
−∞ du f (u, v0)
fv|u(v|u = u0) = f (u0, v)
+∞
−∞ dv f (v, u0)
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Let u and v be two Cartesian parameters (then, volumetric probabilities and probability densities are identical). Given a probability density f (u, v) , one defines the two marginal probability densities fu(u) =
+∞
−∞ dv f (u, v)
fv(v) =
+∞
−∞ du f (u, v)
and the two conditional probability densities fu|v(u|v0) = f (u, v0)
+∞
−∞ du f (u, v0)
fv|u(v|u0) = f (u0, v)
+∞
−∞ dv f (v, u0)
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Let u and v be two Cartesian parameters (then, volumetric probabilities and probability densities are identical). Given a probability density f (u, v) , one defines the two marginal probability densities fu(u) =
+∞
−∞ dv f (u, v)
fv(v) =
+∞
−∞ du f (u, v)
and the two conditional probability densities fu|v(u|v) = f (u, v)
+∞
−∞ du f (u, v)
fv|u(v|u) = f (u, v)
+∞
−∞ dv f (v, u)
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One has fu|v(u|v) = f (u, v) fv(v) fv|u(v|u) = f (u, v) fu(u) from where (a joint distribution can be expressed by a condi- tional distribution times a marginal distribution) f (u, v) = fu|v(u|v) fv(v) = fv|u(v|u) fu(u) from where (Bayes theorem) fu|v(u|v) = fv|u(v|u) fu(u) fv(v)
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Recall: f (u, v) = fu|v(u|v) fv(v) = fv|u(v|u) fu(u) . The two quantities u and v are said to have independent un- certainties if, in fact, f (u, v) = fu(u) fv(v) (the joint distribution equals the product of the two marginal distributions). This implies (and is implied by) fu|v(u|v) = fu(u) ; fv|u(v|u) = fv(v) .
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5 10 15 20
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two quantities with independent uncertainties (the joint distribution is the product
- f the two marginal distributions)
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Let u and v be two Cartesian parameters (then, volumet- ric probabilities and probability densities are identical). Let f (u, v) , be a probability density that is not qualitatively differ- ent from a two-dimensional Gaussian. The mean values are u =
+∞
−∞ du
+∞
−∞ dv u f (u, v)
v =
+∞
−∞ du
+∞
−∞ dv v f (u, v)
the variances are cuu = σ2
u =
+∞
−∞ du
+∞
−∞ dv (u − u)2 f (u, v)
cvv = σ2
v =
+∞
−∞ du
+∞
−∞ dv (v − v)2 f (u, v)
and the covariance is cuv =
+∞
−∞ du
+∞
−∞ dv (u − u)(v − v) f (u, v)
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The covariance matrix is C =
- cuu
cuv cvu cvv
- =
- σ2
u
cuv cvu σ2
v
- .
It is symmetric and positive definite (or, at least, non-negative). Note: the correlation, defined as ρuv = cuv σu σv
=
cuv
√cuu √cvv
, has the property
−1 ≤ ρuv ≤ +1 .
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The general form of a covariance matrix is C = c11 c12 c13 . . . c21 c22 c23 . . . c31 c32 c33 . . . . . . . . . . . . ...
=
σ2
1
c12 c13 . . . c21 σ2
2
c23 . . . c31 c32 σ2
3
. . . . . . . . . . . . ... . The quantities with immediate interpretation are the standard deviations
{σ1, σ2, σ3, . . . }
and the correlation matrix R = 1 ρ12 ρ13 . . . ρ21 1 ρ23 . . . ρ31 ρ32 1 . . . . . . . . . . . . ...
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