Information geometry

Introduction

The following introduction is based on Methods of Information Geometry.

Information and probability

Define an n-set to be a set V with cardinality | V | = n . To choose an element v (value, state, point, outcome) from an n-set V, one needs to specify log b ⁡ n b-sets (default b=2), if one disregards all but the cardinality. That is, I ( v ) = log ⁡ n nats of information are required to specify v; equivalently, I ( v ) = log 2 ⁡ n bits are needed.

By considering the occurrences C of values from V , one has an alternate way to refer to v ∈ V , through C . First, one chooses an occurrence c ∈ C , which requires information of I ( c ) = log 2 ⁡ | C | bits. To specify v, one subtracts the excess information used to choose one c from all those linked to v , this is I ( c v ) = log 2 ⁡ | C v | . Then, | C | | C v | is the number of | C v | portions fitting into | C | . Thus, one needs I ( v ) = log 2 ⁡ | C | | C v | bits to choose one of them. So the information (variable size, code length, number of bits) needed to refer to v , considering its occurrences in a message is

I ( v ) = − log 2 ⁡ p ( v )

Finally, p ( v ) I ( v ) is the normalized portion of information needed to code all occurrences of one v . The averaged code length over all values is H ( V ) = − ∑ p ( v ) log ⁡ p ( v ) . H ( V ) is called the entropy of a random variable V .

Statistical model, Parameters

With a probability distribution p one looks at a variable V through an observation context like a message or an experimental setup.

The context can often be identified by a set of parameters through combinatorial reasoning. The parameters can have an arbitrary number of dimensions and can be very local or less so, as long as the context given by a certain ξ = [ ξ i ] ∈ R n produces every value of V , i.e. the support supp ( V ) does not change as function of ξ . Every ξ determines one probability distribution for V . Basically all distributions for which there exists an explicit analytical formula fall into this category (Binomial, Normal, Poisson, ...). The parameters in these cases have a concrete meaning in the underlying setup, which is a statistical model for the context of V .

The parameters are quite different in nature from V itself, because they do not describe V , but the observation context for V .

A parameterization of the form

p ( v ) = ∑ ξ i p i ( v ) = ξ i p i

with

∑ p i ( v j ) = 1 and ∑ ξ i = 1 ,

that mixes different distributions p i ( v ) , is called a mixture distribution, mixture or m -parameterization or mixture for short. All such parameterizations are related through an affine transformation ρ = A ξ + B . A parameterization with such a transformation rule is called flat.

A flat parameterization for I ( v ) = log ⁡ p ( v ) = E ( v ) + ∑ ξ i F i ( v ) is an exponential or e parameterization, because the parameters are in the exponent of p ( v ) . There are several important distributions, like Normal and Poisson, that fall into this category. These distributions are collectively referred to as exponential family or e -family. The p -manifold for such distributions is not affine, but the log ⁡ p manifold is. This is called e -affine. The parameterization log ⁡ p ( v ) = E ( v ) + ∑ ξ i F i ( v ) − ψ ( ξ ) for the exponential family can be mapped to the one above by making ψ ( ξ ) another parameter and extend [ F i ] → [ F i , 1 ] .

Differential geometry applied to probability

In information geometry, the methods of differential geometry are applied to describe the space of probability distributions for one variable V . This is done by using a coordinate or atlas ξ ∈ R n . Furthermore, the probability p ( v ; ξ ) must be a differentiable and invertible function of ξ . In this case, the [ ξ i ] are coordinates of the p ( v ; ξ ) -space, and the latter is a differential manifold M .

Derivatives are defined as is usual for a differentiable manifold:

∂ i f = ∂ f ∂ ξ i := ∂ f ¯ ∂ ξ i

with f ¯ = f ∘ ξ − 1 , for f ∈ F ( M ) a real-valued function on M .

Given a function f on M , one may "geometrize" it by taking it to define a new manifold. This is done by defining coordinate functions on this new manifold as

ϕ = ( f ∘ ξ − 1 ) − 1 = ξ ∘ f − 1 .

In this way one "geometricizes" a function f , by encoding it into the coordinates used to describe the system.

For f = log the inverse is f − 1 = exp and the resulting manifold of log ⁡ p points is called the e -representation. The p manifold itself is called the m -representation. The e - or m -representations, in the sense used here, does not refer to the parameterization families of the distribution.

Tangent space

In standard differential geometry, the tangent space on a manifold M at a point q is given by:

T q M = { X i ∂ i | X ∈ R n , ∂ i = ∂ ∂ ξ i }

In ordinary differential geometry, there is no canonical coordinate system on the manifold; thus, typically, all discussion must be with regard to an atlas, that is, with regard to functions on the manifold. As a result, tangent spaces and vectors are defined as operators acting on this space of functions. So, for example, in ordinary differential geometry, the basis vectors of the tangent space are the operators ∂ i .

However, with probability distributions p ( v ; ξ ) , one can calculate value-wise. So it is possible to express a tangent space vector directly as X i ∂ i p ( m -representation ) or X i ∂ i log ⁡ p ( e -representation ), and not as operators.

alpha representation

Important functions f of p are coded by a parameter α with the important values 1 , 0 and − 1 :

mixed or m -representation ( α = − 1 ): ℓ ( − 1 ) = 2 1 − α p 1 − α 2 = p

exponential or e -representation ( α = 1 ): ℓ = ℓ ( 1 ) = log ⁡ p ( X ( e ) = 1 p X ( m ) )

0 -representation ( α = 0 ): ℓ ( 0 ) = 2 1 − α p 1 − α 2 = 2 p ( X ( 0 ) = 1 p X ( m ) )

Distributions that allow a flat parameterization ℓ α ( v ; ξ ) = E ( v ) + ξ i F i ( v ) are called collectively α -family ( m -, e - or 0 -family ) of distributions and the according manifold is called α -affine.

The α tangent vector is X ( α ) = X i ∂ i ℓ α .

Inner product

One may introduce an inner product on the tangent space of manifold M at point q as a linear, symmetric and positive definite map

⟨ , ⟩ q : T q × T q → R .

This allows a Riemannian metric to be defined; the resulting manifold is a Riemannian manifold. All of the usual concepts of ordinary differential geometry carry over, including the norm

| | X | | = ⟨ X , X ⟩ ,

the line element d s , the volume element d V , and the cotangent space

T q ∗ M = { T q → R }

that is, the dual space to the tangent space T q . From these, one may construct tensors, as usual.

Fisher metric as inner product

For probability manifolds such an inner product is given by the Fisher information metric.

Here are equivalent formulas of the Fisher information metric.

g i j = ∑ p ∂ i ℓ ∂ j ℓ = E ( ∂ i ℓ ∂ j ℓ )
∂ i ℓ , the i base vector in the e -representation, is also called the score.

g i j = − E ( ∂ i ∂ j ℓ ) ,
because ∂ j ∑ p ∂ i ℓ = ∑ ( ∂ j p ∂ i ℓ + p ∂ i ∂ j ℓ ) = ∂ j ∂ i ∑ p = 0

g i j = ∑ 1 p ∂ i p ∂ j p

g i j α = ∑ ∂ i ℓ ( α ) ∂ j ℓ ( − α ) . This is the same for ± 1 and 0 families.

g i j = D [ ∂ i ∂ j | | ] = D [ | | ∂ i ∂ j ] = − D [ ∂ i | | ∂ j ]
D ( p | | q ) ≥ 0 with minimum 0 for p = q entails D [ ∂ i | | ] = ∂ i D ( p | | p ) = 0 and D [ | | ∂ j ] = ∂ j ′ D ( p | | p ) = 0
∂ i is applied only to the first parameter, and ∂ i ′ only to the second.
D ( p | | q ) = D ( − 1 ) ( p | | q ) = D ( 1 ) ( q | | p ) = ∑ p log ⁡ p q is the Kullback-Leibler divergence or relative entropy applicable to the ± 1 -families.
g i j = − D [ ∂ i | | ∂ j ] = − ∂ j ′ ∂ i ∑ p ( log ⁡ p − log ; q ) = ∑ ∂ i p ∂ j q q = [ p = q ] = ∑ p ∂ i ℓ ∂ j ℓ
For α ≠ ± 1 one has D ( α ) = 4 1 − α 2 ( 1 − ∑ p 1 − α 2 q 1 + α 2 ) .
D ( 0 ) ( p | | q ) = 2 ∑ ( ( p ) − ( q ) ) 2 = 4 ( 1 − ∑ p q ) is the Hellinger distance applicable to the 0 -family. − D ( 0 ) [ ∂ i | | ∂ j ] also evaluates to the Fisher metric.

This relation with a divergence D ( p | | q ) will be revisited further down.

The Fisher metric is motivated by

it satisfying the requirements for an inner product

its invariance for a sufficient statistic deterministic mapping from one variable to another and more general G Y + G ( Y | X ) = G X for p ( y ) = p ( y | x ) p ( x ) , i.e. a broadened distribution has smaller G = g i j .

it being the Cramér–Rao bound.
E [ X ( e ) ] = E [ X i ∂ i log ⁡ p ] = 0 , therefore any B ∈ R | V | satisfying E [ B ] = 0 belongs to T p ( e ) .
For any A ∈ R | V | one has E [ A − E [ A ] ] = 0 , therefore A − E [ A ] ∈ T p ( e ) .
X ( E [ A ] ) = ∑ X ( m ) A = ∑ X i ∂ i A = ∑ X i p ∂ i log ⁡ p A = E [ X ( e ) A ] = E [ X ( e ) A ] − 0 = E [ X ( e ) A ] − E [ X ( e ) E [ A ] ] = E [ X ( e ) ( A − E [ A ] ) ] = E [ X ( e ) Y ( e ) ] = ⟨ X , Y ⟩ .
So Y ( e ) = A − E [ A ] = grad E [ A ] and therefore | | d E [ A ] | | 2 = ⟨ Y ( e ) , Y ( e ) ⟩ = E [ ( A − E [ A ] ) 2 ] = V [ A ] .
| | d E [ A ] | | 2 = G − 1 and with inefficient estimator one gets the Cramér–Rao bound V [ A ] ≥ G − 1 .

Affine connection

Like commonly done on Riemann manifolds, one may define an affine connection (or covariant derivative)

∇ : T M × T M → T M

Given vector fields X and Y lying in the tangent bundle T M , the affine connection ∇ X Y describes how to differentiate the vector field Y along the direction X . It is itself a vector field; it is the sum of the infinitesimal change in the vector field Y , as one moves along the direction X , plus the infinitesimal change of the vector Y due to its parallel transport along the direction X . That is, it takes into account the changing nature of what it means to move a coordinate system in a "parallel" fashion, as one moves about in the manifold. In terms of the basis vectors ∂ k , one has the components:

( ∇ X Y ) k = X i ( ∇ i Y ) k = X i ( ∂ i Y k + Y j Γ i j k )

The Γ i j k are Christoffel symbols. The affine connection may be used for defining curvature and torsion, like is usual in Riemannian geometry.

Alpha connection

A non-metric connection is not determined by a metric tensor g i j ; instead, it is and restricted by the requirement that the parallel transport Π q , q ′ ( ∂ i ) between points q and q ′ must be a linear combination of the base vectors in T q ′ M . Here,

Π q , q ′ ( ∂ j ) = ( ∂ j ) q ′ − d ξ i Γ i j k ( ∂ k ) q ′

expresses the parallel transport of ∂ j as linear combination of the base vectors in T q ′ M , i.e. the new ∂ j minus the change. Note that it is not a tensor (does not transform as a tensor).

For such a metric, one can construct a dual connection ∇ ∗ to make

∂ i g j k = ⟨ ∇ ∂ i ∂ j , ∂ k ⟩ + ⟨ ∂ j , ∇ ∂ i ∗ ∂ k ⟩ = Γ i j , k + Γ i k , j ∗ = 0 ,

for parallel transport using ∇ and ∇ ∗ .

For the mentioned α -families the affine connection is called the α -connection and can also be expressed in more ways.

Γ i j , k ( α ) = E [ ( ∂ i ∂ j ℓ + 1 − α 2 ∂ i ℓ ∂ j ℓ ) ∂ k ℓ ]

Γ i j , k ( α ) = ∑ ∂ i ∂ j ℓ ( α ) ∂ k ℓ ( − α )

Γ i j , k ( α ) = − D ( α ) [ ∂ i ∂ j | | ∂ k ] ( D ( − α ) [ p | | q ] = D ( α ) [ q | | p ] )

For α = ± 1 , 0 :

Γ i j , k ( 0 ) is a metric connection and Γ i j , k ( α ) = Γ i j , k ( 0 ) + α T i j k with T i j k = 1 2 E [ ∂ i l ∂ j l ∂ k l ] .

Γ i j , k ( α ) + Γ i k , j ( − α ) = Γ i j , k ( 0 ) + α T i j k + Γ i k , j ( 0 ) − α T i j k = ∂ i g j k ,

i.e. Γ i j , k ( − α ) is dual to Γ i j , k ( α ) with respect to the Fisher metric.

If Γ i j , k ( α ) = 0 this is called α -affine. Its dual is then − α -affine.

Γ i j , k ( − 1 ) = ∑ ∂ i ∂ j ( ξ i p i ) ∂ k ℓ ( 1 ) = 0 , i.e. 0-affine, and hence Γ i j , k ( 1 ) = 0 , i.e. 1-affine.

Divergence

A function of two distributions (points) D ( p | | q ) ≥ 0 with minimum 0 for p = q entails D [ ∂ i | | ] = ∂ i D ( p | | p ) = 0 and D [ | | ∂ j ] = ∂ j ′ D ( p | | p ) = 0 . ∂ i is applied only to the first parameter, and ∂ i ′ only to the second. ∂ i is the direction, which brought the two points to be equal, when applied to the first parameter, and to diverge again, when applied to the second parameter, i.e. D [ ∂ i | | ] = − D [ | | ∂ i ] . The sign cancels in D [ ∂ i ∂ j | | ] = D [ | | ∂ i ∂ j ] = − D [ ∂ i | | ∂ j ] , which we can define to be a metric g i j = − D [ ∂ i | | ∂ j ] , if always positive.

The absolute derivative of g i j along ∂ i yields candidates for dual connections ∂ i g j k = − D [ ∂ i ∂ j | | ∂ k ] − D [ ∂ j | | ∂ i ∂ k ] = Γ i j , k + Γ i k , j ∗ . This metric and the connections relate to the Taylor series expansion D ( p | | q ) for the first parameter or second parameter. Here for the first parameter:

D [ p | | q ] = 1 2 g i j ( q ) Δ ξ i Δ ξ j + 1 6 h i j k Δ ξ i Δ ξ j Δ ξ k + o ( | | Δ ξ | | 3 ) h i j k = D [ ∂ i ∂ j ∂ k | | ] ∂ i g j k = ∂ i D [ ∂ j ∂ k | | ] = D [ ∂ i ∂ j ∂ k | | ] + D [ ∂ j ∂ k | | ∂ i ] = h i j k − Γ j k , i h i j k = ∂ i g j k + Γ j k , i .

The term D ( p | | q ) is called the divergence or contrast function. A good choice is D ( p | | q ) = ∑ p f ( q p ) with f convex for u > 0 . From Jensen's inequality it follows that D ( p | | q ) ) ≥ f ∑ p q p = f ( 1 ) and, for f ( u ) = u log ⁡ u , we have

D ( p | | q ) = D ( − 1 ) ( p | | q ) = D ( 1 ) ( q | | p ) = ∑ p log ⁡ p q ,

which is the Kullback-Leibler divergence or relative entropy applicable to the ± 1 -families. In the above,

g i j = − D [ ∂ i | | ∂ j ] = − ∂ j ′ ∂ i ∑ p ( log ; p − log ; q ) = ∑ ∂ i p ∂ j q q = ∑ ∂ i p ∂ j log ⁡ q = [ p = q ] = ∑ p ∂ i ℓ ∂ j ℓ

is the Fisher metric. For α ≠ ± 1 a different f yields

D ( α ) = 4 1 − α 2 ( 1 − ∑ p 1 − α 2 p 1 + α 2 ) .

The Hellinger distance applicable to the 0 -family is

D ( 0 ) ( p | | q ) = 2 ∑ ( ( p ) − ( q ) ) 2 = 4 ( 1 − ∑ p q ) .

In this case, − D ( 0 ) [ ∂ i | | ∂ j ] also evaluates to the Fisher metric.

Canonical divergence

We now consider two manifolds S and S ∗ , represented by two sets of coordinate functions [ θ i ] and [ η j ] . The corresponding tangent space basis vectors will be denoted by ∂ i = ∂ ∂ θ i and ∂ i = ∂ ∂ η i . The bilinear map ⟨ , ⟩ : T S × T S ∗ → R associates a quantity ≥ 0 to the dual base vectors. This defines an affine connection ∇ for S and affine connection ∇ ∗ for S ∗ that keep ⟨ X , X ∗ ⟩ constant for parallel transport of X ∈ T S and X ∗ ∈ T S ∗ , defined through ∇ and ∇ ∗ .

If S is flat, then there exists a coordinate system ∂ i , that does not change over S . In order to keep ⟨ ∂ i , ∂ j ⟩ constant, ∂ j must not change either, i.e. S ∗ is also flat. Furthermore, in this case, we can choose coordinate systems such that

⟨ ∂ i , ∂ j ⟩ = δ i j

If S ∗ results as a function f on S , then making η i = θ i ∘ f − 1 , both coordinate system function sets describe S . The connections are such, though, that ∇ makes S flat and ∇ ∗ makes S ∗ flat. This dual space is denoted as ( S , g , ∇ , ∇ ∗ ) .

Because of the linear transform between the flat coordinate systems, we have ∂ j = ( ∂ j θ i ) ∂ i = g i j ∂ i and ∂ i = ( ∂ i η j ) ∂ j = g i j ∂ j .

Because ∂ j θ i = ∂ i θ j and so for η it is possible to define two potentials ψ ( θ ) and ϕ ( η ) through ∂ i ψ = η i and ∂ i ϕ = θ i ( Legendre transform ).These are ψ ( θ ) = max η { θ i η i − ϕ ( η ) } and ϕ ( η ) = max θ { θ i η i − ψ ( θ ) } .

Then

This naturally leads to the following definition of a canonical divergence:

D ( p | | q ) = ψ ( p ) + ϕ ( q ) − θ i ( p ) η i ( q )

Note the summation that is a representation of the metric due to ⟨ ∂ i , ∂ j ⟩ = δ i j .

Properties of divergence

The meaning of the canonical divergence depends on the meaning of the metric ⟨ ∂ i , ∂ j ⟩ = δ i j and vice versa ( g i j = − D [ ∂ i | | ∂ j ] ). For the α = ± 1 metric (Fisher metric) with the dual connections this is the relative entropy. For the self-dual Euclidean space ψ = ϕ = 1 2 ∑ ( θ i ) 2 leads to D ( p | | q ) = 1 2 ∑ ( θ i ( p ) − θ i ( q ) ) 2 = 1 2 d ( p , q ) 2

Similar to the Euclidean space the following holds:

Triangular relation: D ( p | | q ) + D ( q | | r ) − D ( p | | r ) = ( θ i ( p ) − θ i ( q ) ) ( η i ( r ) − η i ( q ) ) (just substitute ϕ ( η ) = θ i η i − ψ ( θ ) )
If ( g , ∇ , ∇ ∗ ) is not dually flat then this generalizes to:
D ( p | | q ) + D ( q | | r ) − D ( p | | r ) = ⟨ E − 1 ( p ) , E − 1 ( r ) ⟩ + o ( max { | | ξ ( p ) − ξ ( q ) | | , | | ξ ( p ) − ξ ( r ) | | } 3 )
The last part drops in case of dual flatness. E is the exponential map.

Pythagorean Theorem: For p and r meeting on orthogonal lines at q ( ( θ i ( p ) − θ i ( q ) ) ( η i ( r ) − η i ( q ) ) = 0 )
D ( p | | r ) = D ( p | | q ) + D ( q | | r )
For q ∈ S and p , r ∈ M with M a ∇ ∗ -autoparallel sub-manifold D ( p | | q ) = min D ( p | | r ) implies that the ∇ -geodesic connecting p and q is orthogonal to M .

By projecting ( g , ∇ , ∇ ∗ ) onto ( g γ , ∇ γ , ∇ γ ∗ ) of a curve γ : [ a , b ] → S one can calculate
the divergence of the curve D γ ( γ ( b ) | | γ ( a ) ) = ∫ ∫ g γ ( s ) μ ( t ) μ ( s ) d s d t where g γ = g i j γ ˙ i γ ˙ j
and μ ( t ) = e ∫ a t Γ γ ( s ) d s with Γ γ ( s ) = { γ ˙ i γ ˙ j Γ i j , k + γ ¨ j g i j } γ ˙ k / g γ .
With Γ γ ( s ) = 0 this becomes D γ ( γ ( b ) | | γ ( a ) ) = ∫ a b ( b − s ) g γ ( s ) d s .

For an autoparallel sub-manifold parallel transport in it can be expressed with the sub-manifold's base vectors, i.e. ∇ ∂ a ∂ b = Γ a b c ∂ c . A one-dimensional autoparallel sub-manifold is a geodesic.

Canonical divergence for the exponential family

For the exponential family p ( v ; θ ) = exp ⁡ [ C ( v ) + θ i F i ( v ) − ψ ( θ ) ] one has exp ⁡ [ ψ ( θ ) ] = ∑ ( C ( v ) + θ i F i ) . Applying ∂ i on both sides yields η i ( θ ) = ∂ i ψ ( θ ) = ∑ F i ( v ) p ( v ; θ ) = E [ F i ] . The other potential ϕ ( θ ) = θ i η i ( θ ) − ψ ( θ ) = θ i ∑ p F i − ψ = E [ log ; p − C ] = − H ( p ) − E [ C ] ( H is entropy, θ i F i = log ⁡ p − C ( v ) + ψ ( θ ) and E [ ψ ( θ ) ] = ψ ( θ ) was used). g i j = E [ ∂ i ℓ ∂ j ℓ ] = E [ ( F i − η i ) ( F j − η j ) ] = V [ η ] is the covariance of η , the Cramér–Rao bound, i.e. an efficient estimator must be exponential.

The canonical divergence is given by the Kullback-Leibler divergence D ( p | | q ) = ∑ p ( log ⁡ p − log ⁡ q ) and the triangulation is D ( p | | q ) + D ( q | | r ) − D ( p | | r ) = ∑ ( p − q ) ( log ; p − log ; q ) .

The minimal divergence to a sub-manifold given by a restriction like some constant η i means maximizing H ( p ) + E [ C ] . With C = 0 this corresponds to the maximum entropy principle.

Canonical divergence for general alpha families

For general α -affine manifolds with ℓ ( α ) = C ( v ) + θ i F i one has:

η i = ∑ F i ℓ ( − α ) ∂ j η i = g i j = ∑ ∂ i ℓ ( α ) ∂ j ℓ ( − α ) = ∑ F i ∂ j ℓ ( − α ) Ψ ( α ≠ − 1 ) ( θ ) = 2 1 + α ∑ p Ψ ( α = − 1 ) ( θ ) = ∑ p ( log ⁡ p − 1 ) ψ ( θ ) = Ψ ( α ) ϕ ( θ ) = Ψ ( − α ) − ∑ C ( x ) ℓ ( − α ) D α ( p | | q ) = Ψ ( α ) + Ψ ( − α ) − ∑ ℓ p ( α ) ℓ q ( − α ) D α ≠ ± 1 ( p | | q ) = 4 1 − α 2 ∑ { 1 − α 2 p + 1 + α 2 q − p 1 − α 2 q 1 + α 2 } D α = ± 1 ( p | | q ) = ∑ { p − q + p log ⁡ p q } θ i η i ′ = ∑ { ℓ ( α ) ( v ; θ ) − C ( v ) } ℓ ( − α ) ( v ; θ ′ ) D ( θ | | θ ′ ) = ψ ( θ ) + ϕ ( θ ) − θ i η i ′

The connection induced by the divergence is not flat unless α = ± 1 . Then the Pythagorean theorem for two curves intersecting orthogonally at q is:

D ( α ) ( p | | r ) = D ( α ) ( p | | q ) + D ( α ) ( q | | r ) − 1 − α 2 4 D ( α ) ( p | | q ) D ( α ) ( q | | r )

History

The history of information geometry is associated with the discoveries of at least the following people, and many others

Sir Ronald Aylmer Fisher

Harald Cramér

Calyampudi Radhakrishna Rao

Harold Jeffreys

Solomon Kullback

Jean-Louis Koszul

Richard Leibler

Claude Shannon

Imre Csiszár

N. N. Cencov (also written as Chentsov)

Bradley Efron

Paul Vos

Shun'ichi Amari

Hiroshi Nagaoka

Robert Kass

Shinto Eguchi

Ole Barndorff-Nielsen

Frank Nielsen

Giovanni Pistone

Bernard Hanzon

Damiano Brigo

Applications

Information geometry can be applied where parametrized distributions play a role.

Here an incomplete list:

statistical inference

time series and linear systems

quantum systems

neural networks

machine learning

statistical mechanics

biology

statistics

mathematical finance