The probability simplex

Manifolds.FisherRaoMetricType
FisherRaoMetric <: AbstractMetric

The Fisher-Rao metric or Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.

See for example the ProbabilitySimplex.

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Manifolds.ProbabilitySimplexType
ProbabilitySimplex{n} <: AbstractEmbeddedManifold{ℝ,DefaultEmbeddingType}

The (relative interior of) the probability simplex is the set

\[Δ^n := \biggl\{ p ∈ ℝ^{n+1}\ \big|\ p_i > 0 \text{ for all } i=1,…,n+1, \text{ and } ⟨\mathbb{1},p⟩ = \sum_{i=1}^{n+1} p_i = 1\biggr\},\]

where $\mathbb{1}=(1,…,1)^{\mathrm{T}}∈ ℝ^{n+1}$ denotes the vector containing only ones.

This set is also called the unit simplex or standard simplex.

The tangent space is given by

\[T_pΔ^n = \biggl\{ X ∈ ℝ^{n+1}\ \big|\ ⟨\mathbb{1},X⟩ = \sum_{i=1}^{n+1} X_i = 0 \biggr\}\]

The manifold is implemented assuming the Fisher-Rao metric for the multinomial distribution, which is equivalent to the induced metric from isometrically embedding the probability simplex in the $n$-sphere of radius 2. The corresponding diffeomorphism $\varphi: \mathbb Δ^n → \mathcal N$, where $\mathcal N \subset 2𝕊^n$ is given by $\varphi(p) = 2\sqrt{p}$.

This implementation follows the notation in [ÅströmPetraSchmitzerSchnörr2017].

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Base.expMethod
exp(M::ProbabilitySimplex,p,X)

Compute the exponential map on the probability simplex.

\[\exp_pX = \frac{1}{2}\Bigl(p+\frac{X_p^2}{\lVert X_p \rVert^2}\Bigr) + \frac{1}{2}\Bigl(p - \frac{X_p^2}{\lVert X_p \rVert^2}\Bigr)\cos(\lVert X_p\rVert) + \frac{1}{\lVert X_p \rVert}\sqrt{p}\sin(\lVert X_p\rVert),\]

where $X_p = \frac{X}{\sqrt{p}}$, with its division meant elementwise, as well as for the operations $X_p^2$ and $\sqrt{p}$.

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Base.logMethod
log(M::ProbabilitySimplex, p, q)

Compute the logarithmic map of p and q on the ProbabilitySimplex M.

\[\log_pq = \frac{d_{Δ^n}(p,q)}{\sqrt{1-⟨\sqrt{p},\sqrt{q}⟩}}(\sqrt{pq} - ⟨\sqrt{p},\sqrt{q}⟩p),\]

where $pq$ and $\sqrt{p}$ is meant elementwise.

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Manifolds.change_metricMethod
change_metric(M::ProbabilitySimplex, ::EuclideanMetric, p, X)

To change the metric, we are looking for a function $c\colon T_pΔ^n \to T_pΔ^n$ such that for all $X,Y ∈ T_pΔ^n$

\[ ⟨X,Y⟩ = X^\mathrm{T}Y = \sum_{i=1}^{n+1}\frac{c(X)_ic(Y)_i}{p_i} = g_p(X,Y)\]

and hence $C(X)_i = X_i\sqrt{p_i}, i=1,…,n+1$.

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Manifolds.change_representerMethod
change_representer(M::ProbabilitySimplex, ::EuclideanMetric, p, X)

Given a tangent vector with respect to the metric from the embedding, the EuclideanMetric, the representer of a linear functional on the tangent space is adapted as $Z = p .* X$, since this “compensates” for the divsion by $p$ in the Riemannian metric on the ProbabilitySimplex.

To be precise for any $Y ∈ T_pΔ^n$ we are looking for $Z ∈ T_pΔ^n$ such that

\[ ⟨X,Y⟩ = X^\mathrm{T}Y = \sum_{i=1}^{n+1}\frac{Z_iY_i}{p_i} = g_p(Z,Y)\]

and hence $Z_i = X_ip_i, i=1,…,n+1$.

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ManifoldsBase.check_pointMethod
check_point(M::ProbabilitySimplex, p; kwargs...)

Check whether p is a valid point on the ProbabilitySimplex M, i.e. is a point in the embedding with positive entries that sum to one The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.check_vectorMethod
check_vector(M::ProbabilitySimplex, p, X; kwargs... )

Check whether X is a tangent vector to p on the ProbabilitySimplex M, i.e. after check_point(M,p), X has to be of same dimension as p and its elements have to sum to one. The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.innerMethod
inner(M::ProbabilitySimplex,p,X,Y)

Compute the inner product of two tangent vectors X, Y from the tangent space $T_pΔ^n$ at p. The formula reads

\[g_p(X,Y) = \sum_{i=1}^{n+1}\frac{X_iY_i}{p_i}\]

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ManifoldsBase.inverse_retractMethod
inverse_retract(M::ProbabilitySimplex, p, q, ::SoftmaxInverseRetraction)

Compute a first order approximation by projection. The formula reads

\[\operatorname{retr}^{-1}_p q = \bigl( I_{n+1} - \frac{1}{n}\mathbb{1}^{n+1,n+1} \bigr)(\log(q)-\log(p))\]

where $\mathbb{1}^{m,n}$ is the size (m,n) matrix containing ones, and $\log$ is applied elementwise.

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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::ProbabilitySimplex{n})

Returns the manifold dimension of the probability simplex in $ℝ^{n+1}$, i.e.

\[ \dim_{Δ^n} = n.\]

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ManifoldsBase.projectMethod
project(M::ProbabilitySimplex, p, Y)

project Y from the embedding onto the tangent space at p on the ProbabilitySimplex M. The formula reads

\[\operatorname{proj}_{Δ^n}(p,Y) = Y - ⟨\mathbb 1,Y⟩p,\]

where $\mathbb 1 ∈ ℝ$ denotes the vector of ones.

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ManifoldsBase.projectMethod
project(M::ProbabilitySimplex, p)

project p from the embedding onto the ProbabilitySimplex M. The formula reads

\[\operatorname{proj}_{Δ^n}(p) = \frac{1}{⟨\mathbb 1,p⟩}p,\]

where $\mathbb 1 ∈ ℝ$ denotes the vector of ones. Not that this projection is only well-defined if $p$ has positive entries.

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ManifoldsBase.representation_sizeMethod
representation_size(::ProbabilitySimplex{n})

return the representation size of points in the $n$-dimensional probability simplex, i.e. an array size of (n+1,).

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ManifoldsBase.retractMethod
retract(M::ProbabilitySimplex, p, X, ::SoftmaxRetraction)

Compute a first order approximation by applying the softmax function. The formula reads

\[\operatorname{retr}_p X = \frac{p\mathrm{e}^X}{⟨p,\mathrm{e}^X⟩},\]

where multiplication, exponentiation and division are meant elementwise.

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Literature