We showed [3] that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology; here we use Mathematica to illustrate these neighbourhoods. Using Mathematica, without which the computations would have been prohibitively tedious, we derived also the information geometry of the 3-manifold of Mckay bivariate gamma distributions, which can provide a metrization of departures from randomness and independence for bivariate processes.  As in the case of bivariate normal manifolds, we have negative scalar curvature but here it is not constant; details are provided in our paper [3].

1  Gamma distributions and randomness

The family of gamma probability density functions is given by

\(TraditionalForm\`{p(
          x; β, α) = \(\((α\/β)\)\^α\) \(x\^\(α - 1\)\/\(Γ(α)\)\) e\^\(\(-\(α\/β\)\) x\) | α, β\ ϵ\ \(ℝ\^+\)}, \ \ \(\(x\)\(\ \)\(ϵ\)\(\ \)\(ℝ\^+\)\(\ \ \ \ \ \ \ \ \ \ \ \)\)\)

so the space of parameters is topologically \(TraditionalForm\`\(ℝ\^+\)×\ \(ℝ\^+\)\).  It is an exponential family and it includes as a special case (α=1) the exponential distribution itself, which complements the Poisson process on a line.  The property of having sample standard deviation independent of the mean actually characterizes gamma distributions, as shown recently by Hwang and Hu [10].  Of course, the exponential distribution has unit coefficient of variation.
The information geometry is known for the gamma family (1) and has been applied recently to represent and metrize departures from randomness of, for example, the processes that allocate gaps between occurrences of each amino acid along a protein chain within the Saccharomyces cerevisiae genome, see Cai et al [4], clustering of galaxies and communications, Dodson [5,7,6].

Proposition 1.1 [3]  Every neighbourhood of an exponential distribution contains a neighbourhood of gamma  distributions, using the subspace topology of (ℝ^3)and information theoretic immersions.

Outline Proof: Dodson and Matsuzoe[8] have provided an affine immersion in Euclidean (ℝ^3) for 𝒢  the manifold of gamma distributions with Fisher information metric. The coordinates  \(TraditionalForm\`\((\ θ\^1, \ θ\^2\ )\) = \((μ = α\/β, \ α)\)\)  form a natural coordinate system (cf Amari and Nagaoka [2]) for the gamma manifold 𝒢 . So 𝒢  can be realized in Euclidean (ℝ^3) by the graph of the affine immersion {h,  ξ} where ξ  is the  transversal vector field along h [2, 8]:

h: 𝒢 -> (ℝ^3) :(({{μ}, {α}}))↦(({{μ}, {α}, {\(log\ \(Γ(α)\) - α\ log\ μ\)}})) ,   ξ  = (({{0}, {0}, {1}})) .

The submanifold of exponential distributions is represented by the curve

(0, ∞)-> (ℝ^3): μ ↦{ μ , 1, log (1/μ)}

and a tubular neighbourhood of this curve will contain all immersions for small enough perturbations of exponential distributions.  
This is illustrated in the graphics which show S, an affine embedding of the gamma manifold as a surface in (ℝ^3), and R an (ℝ^3)-tubular neighbourhood of the curve α = 1 in the surface; this curve represents all exponential distributions.

S = 
        ParametricPlot3D[{β, ν, 
            Log[Gamma[ν]] - ν Log[β]}, 
{β, 0.00001, 4}, {ν, 0.0001, 3},  ViewPoint -> {3, 2, 1},  
          BoxRatios -> {1, 1, 1}];
R = 
        ParametricPlot3D[{β - 
              0.6 ((((β^2) + 1))^(-(1/2))) Cos[theta], 
            1 - 0.6 Sin[
                  theta], -0.6 β ((((β^2) + 1))^(-(1/2)))  Cos[theta] - Log[β]}, {β, 0.0001, 
            4}, {theta, 0, 2  Pi}, PlotPoints -> {20, 20}, 
          ViewPoint -> {3, 2, 1}, 
          BoxRatios -> {1, 1, 1}];

Show[S, R];

The value of such topological results lies in the fact that they have qualitative consequences that are stable under small perturbations of a process, something that would be important in real applications.  It gives confidence in the use of gamma distributions to model near random processes. For comparison purposes, we recall that the information geometry of the univariate and  multivariate normal distributions also are known, see for example Sato et al [11] for the bivariate case.  We note in particular that the univariate and bivariate normal distributions have constant negative scalar curvature, so geometrically they constitute parts of pseudospheres.

2  Geometry of the Mckay bivariate gamma 3-manifold

The classical family of Mckay bivariate gamma probability density functions defined on  0 < x < y < ∞ with parameters (Subscript[α, 1]), (Subscript[σ, 12]), (Subscript[α, 2]) > 0, and  (Subscript[σ, 12]) the covariance of X and Y , is given by:

f(x_, 
          y_) = (((((((Subscript[α, 1])/(Subscript[σ, 12]))))^(((((Subscript[α, 1]) + (Subscript[α, 2])))/2))) (x^((Subscript[α, 1]) - 1)) (((y - x))^((Subscript[α, 2]) - 1)) (E^(-y (Sqrt[((Subscript[α, 1])/(Subscript[σ, 12]))]))))/(Γ((Subscript[α, 1])) Γ((Subscript[α, 2]))));

We denote the set of such pdfs by M. The marginal distributions of X and Y are as follows.

{(Subscript[f, X])(x)  = 
        FullSimplify[(∫_x^∞) f(x, 
                y) \[DifferentialD]y, {y > x > 0, (Subscript[α, 1]) > 
              0, (Subscript[σ, 12]) > 0, (Subscript[α, 2]) > 0}], (Subscript[f, Y])(y) = 
        FullSimplify[(∫_0^y) f(x, y) \[DifferentialD]x, {y > 
              x > 0, (Subscript[α, 1]) > 0, (Subscript[σ, 12]) > 0, (Subscript[α, 2]) > 
              0}]      }

Using standard procedures from [1,2],  M becomes  a Riemannian  3-manifold with Fisher information metric G = [(Subscript[g, ij])] obtained from expectations of the second derivatives of the log-likelihood function as follows:

Outer[FullSimplify[(∫_0^∞)(∫_x^∞)-f(x, y)Simplify[(((∂^2) log(
                                f(x, y)))/(∂#1 ∂#2))]\[DifferentialD]y\[DifferentialD]x,{y > x > 0,  (Subscript[α, 1]) > 0, (Subscript[σ, 12]) > 
                      0, (Subscript[α, 2]) > 0}]&,{(Subscript[α, 1]), (Subscript[σ, 12]), (Subscript[α, 2])},{(Subscript[α, 1]), (Subscript[σ, 12]), (Subscript[α, 2])}]//MatrixForm

The inverse matrix is  [(g^ij)] = (([(Subscript[g, ij])])^(-1)) and then theChristoffel symbols are defined by  \(Γ\&i\_jk = ∑\+\(h = 1\)\%3\ \(1\/2\) \(\(\(\ \)\(g\)\)\&ih\) \((∂\_x\&k\ g\_jh + ∂\_x\&j\ 
                        g\_kh - ∂\_x\&h\ g\_jk)\)\); (i, j, k = 1, 2, 3). The input is illustrated by:

(Subscript[(Overscript[Γ, 1]), 11]) = (1/2) ( (Overscript[g, 11])((Overscript[(∂_x), 1]) (Subscript[g, 11]) + (Overscript[(∂_x), 1]) (Subscript[g, 11]) - (Overscript[(∂_x), 1]) 
                      (Subscript[g, 11])) + (Overscript[g, 12])((Overscript[(∂_x), 1]) 
                      (Subscript[g, 21]) + (Overscript[(∂_x), 1]) (Subscript[g, 21]) - (Overscript[(∂_x), 2]) 
                      (Subscript[g, 11])) + (Overscript[g, 13])((Overscript[(∂_x), 1]) 
                      (Subscript[g, 31]) + (Overscript[(∂_x), 1]) (Subscript[g, 31]) - (Overscript[(∂_x), 3]) 
                      (Subscript[g, 11]))) // PowerExpand // FullSimplify

The components of the curvature tensor  \(R\_ijkm = ∑\+\(h = 1\)\%3\( 
                      g\_hm\)(∂\_x\&i\ Γ\&h\_jk - ∂\_x\&j\ Γ\&h\_ik + Γ\&h\_im\ Γ\&m\_jk - Γ\&h\_jm\ Γ\&m\_ik)\); (i, j, k = 1, 2, 3)   are given by, eg:

(Subscript[R, 1212])=(Subscript[g, 12])  ((Overscript[(∂_x), 1]) (Subscript[(Overscript[Γ, 1]), 21]) - (Overscript[(∂_x), 2]) (Subscript[(Overscript[Γ, 1]), 11]) + (Subscript[(Overscript[Γ, 1]), 11]) (Subscript[(Overscript[Γ, 1]), 21]) - (Subscript[(Overscript[Γ, 1]), 21]) (Subscript[(Overscript[Γ, 1]), 11]) + (Subscript[(Overscript[Γ, 1]), 12]) (Subscript[(Overscript[Γ, 2]), 21]) - (Subscript[(Overscript[Γ, 1]), 22]) (Subscript[(Overscript[Γ, 2]), 11]) + (Subscript[(Overscript[Γ, 1]), 13]) (Subscript[(Overscript[Γ, 3]), 21]) - (Subscript[(Overscript[Γ, 1]), 23]) (Subscript[(Overscript[Γ, 3]), 11])) + 
                (Subscript[g, 22])  ((Overscript[(∂_x), 1]) (Subscript[(Overscript[Γ, 2]), 21]) - (Overscript[(∂_x), 2]) (Subscript[(Overscript[Γ, 2]), 11]) + (Subscript[(Overscript[Γ, 2]), 11]) (Subscript[(Overscript[Γ, 1]), 21]) - (Subscript[(Overscript[Γ, 2]), 21]) (Subscript[(Overscript[Γ, 1]), 11]) + (Subscript[(Overscript[Γ, 2]), 12]) (Subscript[(Overscript[Γ, 2]), 21]) - (Subscript[(Overscript[Γ, 2]), 22]) (Subscript[(Overscript[Γ, 2]), 11]) + (Subscript[(Overscript[Γ, 2]), 13]) (Subscript[(Overscript[Γ, 3]), 21]) - (Subscript[(Overscript[Γ, 2]), 23]) (Subscript[(Overscript[Γ, 3]), 11])) + 
                (Subscript[g, 32])  ((Overscript[(∂_x), 1]) (Subscript[(Overscript[Γ, 3]), 21]) - (Overscript[(∂_x), 2]) (Subscript[(Overscript[Γ, 3]), 11]) + (Subscript[(Overscript[Γ, 3]), 11]) (Subscript[(Overscript[Γ, 1]), 21]) - (Subscript[(Overscript[Γ, 3]), 21]) (Subscript[(Overscript[Γ, 1]), 11]) + (Subscript[(Overscript[Γ, 3]), 12]) (Subscript[(Overscript[Γ, 2]), 21]) - (Subscript[(Overscript[Γ, 3]), 22]) (Subscript[(Overscript[Γ, 2]), 11]) + (Subscript[(Overscript[Γ, 3]), 13]) (Subscript[(Overscript[Γ, 3]), 21]) - (Subscript[(Overscript[Γ, 3]), 23]) (Subscript[(Overscript[Γ, 3]), 11]))//PowerExpand//FullSimplify

The components of the Ricci curvature are given by the symmetric matrix R=[(Subscript[R, jk])] , where (Subscript[R, jk])  = (Subscript[R, ijkm]) (Overscript[g, im]) , eg:

(Subscript[R, 11]) = (Subscript[R, 1111]) (Overscript[g, 11]) + (Subscript[R, 1112]) (Overscript[g, 12]) + 
            (Subscript[R, 1113]) (Overscript[g, 13]) + (Subscript[R, 2111]) (Overscript[g, 21]) + (Subscript[R, 2112]) (Overscript[g, 22]) + 
            (Subscript[R, 2113]) (Overscript[g, 23]) + (Subscript[R, 3111]) (Overscript[g, 31]) + (Subscript[R, 3112]) (Overscript[g, 32]) + 
            (Subscript[R, 3113]) (Overscript[g, 33]) // PowerExpand // FullSimplify

The scalar curvature R  = (Subscript[R, jk]) (Overscript[g, jk]) is given by:

R = (Subscript[R, 11]) (Overscript[g, 11]) + (Subscript[R, 12]) (Overscript[g, 12]) + (Subscript[R, 13]) 
              (Overscript[g, 13]) + (Subscript[R, 21]) (Overscript[g, 21]) + (Subscript[R, 22]) (Overscript[g, 22]) + (Subscript[R, 23]) 
              (Overscript[g, 23]) + (Subscript[R, 31]) (Overscript[g, 31]) + (Subscript[R, 32]) (Overscript[g, 32]) + (Subscript[R, 33]) (Overscript[g, 33]) // 
          PowerExpand // FullSimplify

4  Applications

The univariate gamma information geometry is known and has been applied recently to represent and metrize departures from randomness of, for example, the processes that allocate gaps between occurrences of each amino acid along a protein chain within the Saccharomyces cerevisiae genome, see Cai et al [4], cosmological void distribution and clustering of galaxies, and communications, Dodson [5,7,6]. Two new case studies are being developed at present, both involve bivariate data from measurements on stochastic porous media. Near-planar stochastic fibrous network structures like, paper and nonwoven textiles, are deposited on a continuous filter bed. Dodson and Sampson [9] used the product of two gamma distributions to obtain the void size distribution, recovering the known random model as a special case.   Some commercial processes involve preferential alignment of fibres in the direction of manufacture, resulting in anisotropic void distributions; then the McKay bivariate model is applicable. The  3-dimensional tomographic images of soil samples in hydrology surveys yield 3-dimensional reconstructions of thestochastic  porous structure and, interestingly,  these generate a bivariate positively correlated pair of random variables   0 < x < y < ∞.   Here, the larger variable represents the size of a void and the smaller variable represents the sizes of the throats that connect it to neighbouring  voids.  This work is being pursued in collaboration with Professor J. Scharkanski of the Federal University of Rio Grande do Sul, Brazil and will be reported elsewhere.
Acknowledgment: The authors are grateful to Peter Mitic for suggesting improvements in the Mathematica formatting.

5 References