# 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{T,boundary} <: AbstractDecoratorManifold{𝔽}

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.

If boundary is set to :open, then the object represents an open simplex. Otherwise, that is when boundary is set to :closed, the boundary is also included:

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

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 [APSS17].

Constructor

ProbabilitySimplex(n::Int; boundary::Symbol=:open)
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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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Base.randMethod
rand(::ProbabilitySimplex; vector_at=nothing, σ::Real=1.0)

When vector_at is nothing, return a random (uniform over the Fisher-Rao metric; that is, uniform with respect to the n-sphere whose positive orthant is mapped to the simplex). point x on the ProbabilitySimplex manifold M according to the isometric embedding into the n-sphere by normalizing the vector length of a sample from a multivariate Gaussian. See [Mar72].

When vector_at is not nothing, return a (Gaussian) random vector from the tangent space $T_{p}\mathrm{\Delta}^n$by shifting a multivariate Gaussian with standard deviation σ to have a zero component sum.

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ManifoldDiff.riemannian_gradientMethod
X = riemannian_gradient(M::ProbabilitySimplex, p, Y)
riemannian_gradient!(M::ProbabilitySimplex, X, p, Y)

Given a gradient $Y = \operatorname{grad} \tilde f(p)$ in the embedding $ℝ^{n+1}$ of the ProbabilitySimplex $Δ^n$, this function computes the Riemannian gradient $X = \operatorname{grad} f(p)$ where $f$ is the function $\tilde f$ restricted to the manifold.

$$$X = p ⊙ Y - ⟨p, Y⟩p,$$$

where $⊙$ denotes the emelementwise product.

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

To change the metric, we are looking for a function $c\colon T_pΔ^n → T_pΔ^n$ such that for all $X,Y ∈ T_pΔ^n$ This can be achieved by rewriting representer change in matrix form as (Diagonal(p) - p * p') * X and taking square root of the matrix

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ManifoldsBase.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 $Z = p .* X .- p .* dot(p, X)$. The first part “compensates” for the division by $p$ in the Riemannian metric on the ProbabilitySimplex and the second part performs appropriate projection to keep the vector tangent.

For details see Proposition 2.3 in [APSS17].

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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

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

When M includes boundary, we can just skip coordinates where $p_i$ is equal to 0, see Proposition 2.1 in [AJLS17].

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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)

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

math \operatorname{proj}_{Δ^n}(p,Y) = Y - \bar{Y} where $\bar{Y}$ denotes mean of $Y$.

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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.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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## Euclidean metric

Base.randMethod
rand(::MetricManifold{ℝ,<:ProbabilitySimplex,<:EuclideanMetric}; vector_at=nothing, σ::Real=1.0)

When vector_at is nothing, return a random (uniform) point x on the ProbabilitySimplex with the Euclidean metric manifold M by normalizing independent exponential draws to unit sum, see [Dev86], Theorems 2.1 and 2.2 on p. 207 and 208, respectively. When vector_at is not nothing, return a (Gaussian) random vector from the tangent space $T_{p}\mathrm{\Delta}^n$by shifting a multivariate Gaussian with standard deviation σ to have a zero component sum.

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## Real probability amplitudes

An isometric embedding of interior of ProbabilitySimplex in positive orthant of the Sphere is established through functions simplex_to_amplitude and amplitude_to_simplex. Some properties extend to the boundary but not all.

This embedding isometrically maps the Fisher-Rao metric on the open probability simplex to the sphere of radius 1 with Euclidean metric. More details can be found in Section 2.2 of [AJLS17].

The name derives from the notion of probability amplitudes in quantum mechanics. They are complex-valued and their squared norm corresponds to probability. This construction restricted to real valued amplitudes results in this embedding.

Manifolds.amplitude_to_simplexMethod
amplitude_to_simplex(M::ProbabilitySimplex, p)

Convert point (real) probability amplitude p on to a point on ProbabilitySimplex. The formula reads $(p_1^2, p_2^2, …, p_{N+1}^2)$. This is an isometry from the interior of the positive orthant of a sphere to interior of the probability simplex.

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Manifolds.simplex_to_amplitudeMethod
simplex_to_amplitude(M::ProbabilitySimplex, p)

Convert point p on ProbabilitySimplex` to (real) probability amplitude. The formula reads $(\sqrt{p_1}, \sqrt{p_2}, …, \sqrt{p_{N+1}})$. This is an isometry from the interior of the probability simplex to the interior of the positive orthant of a sphere.

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## Literature

[AJLS17]
N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer. Information Geometry (Springer Cham, 2017).
[Dev86]
L. Devroye. Non-Uniform Random Variate Generation (Springer New York, NY, 1986).
[Mar72]
G. Marsaglia. Choosing a Point from the Surface of a Sphere. Annals of Mathematical Statistics 43, 645–646 (1972).
[APSS17]
F. Åström, S. Petra, B. Schmitzer and C. Schnörr. Image Labeling by Assignment. Journal of Mathematical Imaging and Vision 58, 211–238 (2017), arXiv:1603.05285.