—
Probabilistic Mappings
of Probability Measures
Albert Tarantola
Université Paris VI & IPGP
Inverse Days, Tahkovuori, Finland, December 2008
—
The Bayes-Popper approach.
Using observations to infer the values of some parameters corresponds to solving an inverse problem. Practitioners sometimes seek the best solution implied by the data, but observations should only be used to falsify possible solutions, not to
deduce any particular solution.
intersection
X
X
Θ-1 Θ
X
Y
intersection
Y
∼
Θ
Y
ϕ
X
Y
ϕ
Y
X
x
X
x → y = ϕ(x)
y
Y
ϕ
Y
X
x
x → y = ϕ(x)
Y
X
X
y
x
x → θx(y)
Y
X
Y
Y
X
X
x
x → θx(y)
Y
Intersection of measures: Given a measure space (Ω, F , µ) ,
let µ1 and µ2 be two measures such that, for every F ∈ F ,
the following expressions make sense:
µ[µ1 , µ2 ][ F ] =
(µ1 ∩ µ2 )[ F ] =
Z
ω∈F
dµ
dµ1
(ω ) 2 (ω ) dµ(ω )
dµ
dµ
µ[µ1 , µ2 ][ F ]
µ[µ1 , µ2 ][Ω]
.
The probability measure (µ1 ∩ µ2 ) is the intersection of the two
measures µ1 and µ2 . The quintuplet {Ω, F , µ, µ1 , µ2 } is a
finite Radon-Nikodym space.
X
intersection
X
X
X
Family of probability measures: Let ( X, F X ) and (Y, FY ) be
two measurable spaces, and assume that to every x ∈ X it is
associated a probability measure θ x on (Y, FY ) . We use the
notation Θ for the set {θ x | x ∈ X } , and we say that Θ is a
family of probability measures from X on (Y, FY ) .
Image: Let ( X, F X ) and (Y, FY ) be two measurable spaces,
Θ = {θ x | x ∈ X } a family of probability measures from X
on (Y, FY ) , and π X a measure on ( X, F X ) . If for any FY ∈
FY , the function θ x [ FY ] is π X -measurable, the image of π X
(by the family Θ ) is the measure on (Y, FY ) , denoted Θ[π X ] ,
defined by the condition
(Θ[π X ])[ FY ] =
Z
X
θ x [ FY ] dπ X ( x )
for every FY ∈ FY
Later on the following notation
d(Θ[π ])(y) =
Z
shall be used for this definition.
X
dθ x (y) dπ ( x )
.
X
X
X
x
x → θx(y)
image of a
probability
distribution
Θ
Y
Y
Reciprocal image: Let ( X, F X , µ X ) and (Y, FY , µY ) be two
measure spaces, πY a measure on (Y, FY ) , and Θ = {θ x | x ∈
X } a family of probability measures from X on (Y, FY ) such
that for every x ∈ X , (Y, FY , µY , θ x , πY ) is a Radon-Nikodym
space. The reciprocal image of πY (by the family Θ ), denoted
Θ-1 [πY ] , is the measure on ( X, F X ) , absolutely continuous
w.r.t. µ X , defined, via its µ X -density, as
d(Θ-1 [πY ])
(x) =
dµ X
Z
Y
(θ x ∩ πY )(y) dµY (y) .
d(Θ-1 [πY ])
( x ) = µY[θ x , πY ][Y ]
dµ X
.
Y
X
x
x → θx(y)
reciprocal image
of a probability
distribution
X
Θ-1
Y
Y
Product of measures: Given two measurable spaces ( X, F X )
and (Y, FY ) , to every pair of measures τX and τY , respectively on ( X, F X ) and on (Y, FY ) , is associated a measure
τX × τY on ( X × Y , F X × FY ) , called the product measure,
defined for every FX ∈ F X and every FY ∈ FY , by
(τX × τY )[ FX × FY ] =
Z
FX
dτX ( x )
Z
FY
dτY (y)
We shall use the notation
d(τX × τY )( x, y) = dτX ( x ) × dτY (y)
.
.
Marginal measures: Given two measurable spaces ( X, F X )
and (Y, FY ) , to every measure π on ( X × Y , F X × FY ) are
associated the two marginal measures (respectively on ( X, F X )
and on (Y, FY ) ) defined (respectively for every FX ∈ F X and
every FY ∈ FY ) by
π X [ FX ] = π [ FX × Y ]
;
πY [ FY ] = π [ X × FY ]
.
Inference space: Given two measure spaces ( X, F X , µ X ) and
(Y, FY , µY ) , given a family Θ = {θ x | x ∈ X } of probability
measures on (Y, FY ) , and given a probability measure π on
( X × Y , F X × FY ) , an inference space is the quintuplet
I = { X × Y , F X × FY , µ X × µY , π , Θ } .
Premise inference space: If an inference space I = { X ×
Y , F X × FY , µ X × µY , π , Θ } is such that
π = π X × πY
,
where π X and πY are the two marginal probability measures
of the probability measure π , if the intersections θ x ∩ πY are
nonempty (for all x ∈ X ), we say that I is a premise inference
space.
Conclusion inference space: If an inference space I = { X ×
Y , F X × FY , µ X × µY , π , Θ } is such that
Θ [ π X ] = πY
,
where π X and πY are the two marginal probability measures
of the probability measure π , we say that I is a conclusion
inference space.
Theorem: If I = { X × Y , F X × FY , µ X × µY , π , Θ } is a
premise inference space, then the quintuplet I = { X × Y ,
 } where the probability measures of
, Θ
F X × FY , µ X × µ Y , π
 = {θx | x ∈ X } are defined as
the family Θ
θx = θ x ∩ πY
 is defined by
and where the probability measure π
 [ FX × FY ] =
π

FX

FY
θx (y) dµY (y) dπ X ( x )
,
is a conclusion inference space, i.e. the two marginal probabil , say π
X and π
Y , are related as
ity measures of π
 [π
X ] = π
Y
Θ
.
Furthermore, one has
eX = π X ∩ Θ-1 [πY ]
π
and
eY = πY ∩ Θ[π X ]
π
.
X
Y
Y
X
Θ-1 Θ
X
Y
intersection
X
Θ-1 Θ
X
Y
X
Y
intersection
Y
intersection
X
X
Θ-1 Θ
X
Y
intersection
Y
∼
Θ
Y
intersection
X
X
Θ-1 Θ
X
Y
intersection
Y
∼
Θ
Y
Theorem: If I = { X × Y , F X × FY , µ X × µY , π , Θ } is a
premise inference space, then the quintuplet I = { X × Y ,
 } where the probability measures of
, Θ
F X × FY , µ X × µ Y , π
 = {θx | x ∈ X } are defined as
the family Θ
θx = θ x ∩ πY
 is defined by
and where the probability measure π
 [ FX × FY ] =
π

FX

FY
θx (y) dµY (y) dπ X ( x )
,
is a conclusion inference space, i.e. the two marginal probabil , say π
X and π
Y , are related as
ity measures of π
 [π
X ] = π
Y
Θ
.
X
x
x → θx(y)
Y
X
x
x → θx(y)
ϕ1
X
x
ϕ2
ϕ3
Y
r1
r2
r3
Y
X
x
x → θx(y)
ϕ1
X
X
ϕ2
Y
r1
r2
x
ϕ3
x
ϕ1 ~
r1
ϕ2 ~
r2
ϕ3 ~
r3
r3
Y
Y
T HE E ND .