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Maxallent: Maximizers of all Entropies and Uncertainty of Uncertainty

The entropy maximum approach (Maxent) was developed as a minimization of the subjective uncertainty measured by the Boltzmann--Gibbs--Shannon entropy. Many new entropies have been invented in the second half of the 20th century. Now there exists a rich choice of entropies for fitting needs. This diversity of entropies gave rise to a Maxent"anarchism". Maxent approach is now the conditional maximization of an appropriate entropy for the evaluation of the probability distribution when our information is partial and incomplete. The rich choice of non-classical entropies causes a new problem: which entropy is better for a given class of applications? We understand entropy as a measure of uncertainty which increases in Markov processes. In this work, we describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). For inference, this approach results in a set of conditionally"most random"distributions. Each distribution from this set is a maximizer of its own entropy. This"uncertainty of uncertainty"is unavoidable in analysis of non-equilibrium systems. Surprisingly, the constructive description of this set of maximizers is possible. Two decomposition theorems for Markov processes provide a tool for this description.


Introduction
Entropy was born in the 19th century as a daughter of energy: dS = δQ/T . Clausius [1], Boltzmann [2] and Gibbs [3] (and others) had developed the physical notion of entropy. At the same time, the famous Boltzmann's formula S = k log W had opened the informational interpretation of entropy. In the 20th century, Hartley [4] and Shannon [5] introduced a logarithmic measure of information in electronic communication in order "to eliminate the psychological factors involved and to establish a measure of information in terms of purely physical quantities" ( [4], p. 536). Information theory is focused on entropy as a measure of uncertainty of subjective choice. This understanding of entropy was returned from information theory to statistical mechanics by Jaynes [6] as a basis of "subjective" statistical mechanics: "Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum entropy estimate. It is least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information. That is to say, when characterizing some unknown events with a statistical model, we should always choose the one that has Maximum Entropy." This is the brief manifesto of the Maxent (maximum of entropy) methodology.
Entropy is used for measurement of uncertainty in a probability distribution. The Maxent method finds the maximally uncertain distribution under given values of some moments. After Jaynes, this approach became very popular in physics [7,8], statistics [9,10], econometrics [11,12] and other disciplines.
The non-classical entropies were invented by Rényi [13] in the middle of the 20th century, simultaneously with the expansion of the Maxent approach. This invention introduced additional uncertainty in the uncertainty evaluation. Maximization of different entropies produces different probability distributions under the same conditions. Now, one has to select the proper entropy functional to use in the Maxent approach. This choice may be non-obvious. The beautiful and transparent understanding of the Maxent distribution as a unique "least biased estimate possible on the given information" is now destroyed by the non-classical entropies. If we consider the non-classical entropies seriously then we have to select the proper entropy for each problem.
If we do not find solid reasons for the entropy selection then we have to accept this "Uncertainty of Uncertainty" (UoU) as the nature of things. In this case, the set of all the Maxent distributions for different entropies will evaluate the unknown "maximally uncertain" distribution under given conditions. We call this method of handling the UoU the "maximization of all entropies" or Maxallent. If there are some reasons for selection of a class of entropy function then we have to select the conditional maximizer of the entropies from this class.
The widest class of entropies we use in this paper are the Csiszár-Morimoto conditional entropies (fdivergencies). They were introduced by Rényi in his famous work [13] where he proposed also the "Rényi entropy". The f -divergencies were studied further by Csiszar [14] and T. Morimoto [15]. For a discrete probability distribution P = (p i ) and the positive "equilibrium distribution" P * = (p * i ), p * i > 0 the general form of the f -divergence is where h(x) is a convex function defined on the open (x > 0) or closed (x ≥ 0) semi-axis. We use here the notation H h (P P * ) to stress the dependence of H h both on p i and p * i . In some practical problems, it is convenient to use a convex function h(x) with singularity at x = 0, for example, h(x) = − ln x (the Burg relative entropy [16]). Therefore, we assume that the function H h (P P * ) is defined for positive P and P * . Convexity of h(x) implies convexity of H h (P P * ) as a function of P . It achieves its minimal value on the equilibrium probability, P = P * (under conditions i p i = 1, and p i > 0). If h(x) is strictly convex then H h (P P * ) is also strictly convex and this minimizer (the equilibrium) is unique.

Maxallent, approach #1: parametrization by monotonic function of one variable
The standard settings for the Maxent approach are: an event space Ω, a divergency H h (P P * ) and a set of moments M r (P ) (r = 1, . . . , k) are given. Here, P is a probability distribution, P * is the "maximally disordered" probability distribution ("equilibrium") and H h (P P * ) measures the deviation of P from P * . Of course, for general probability spaces we have to assume that P is absolutely continuous with respect to P * and that it is possible to compute the divergence H h (P P * ). The Maxent problem is: for given values of the moments M r (P ) (r = 1, . . . , k) find the minimizers of H h (P P * ). That is, on the set of probability distributions with given values of M r (P ) (r = 1, . . . , k) find the distributions that are the closest to the equilibrium P * if we measure the deviation by H h (P P * ). The terminological mess (Maxent and minimizers) appears due to historical reasons. Divergences measure the differences between distributions and we always look for minimizers of them.
To avoid the irrelevant technicalities we consider discrete distributions. Let Ω = {A 1 , A 2 , . . . , A n } be a finite event space with probability distributions P = (p i ). The set of probability distribution is the standard simplex ∆ n−1 in R n . The set of positive distribution (p i > 0) is ∆ n−1 + , the relative interior of the standard simplex.
If rank(m rj ) < k + 1 then just exclude some moments.
The method of Lagrange multipliers gives for P ∈ ∆ n−1 The derivative h ′ is a monotonic function. Let h be strictly convex. Then the inverse function g(y) exists, g(h ′ (x)) = x (for positive x). We can apply the function g to both sides of (2) and write the expression of P and the equations for the Lagrange multipliers λ r that are just the moment conditions j m rj p j = M r : Therefore, for the class of the strictly convex functions h all the positive solutions of the Maxent problem for all f -divergencies are parameterized (3) by the monotonic function g.
The function g should be defined on a real interval (a, b) = h ′ ((0, ∞)) (it might be that a = −∞ or b = ∞). The image of g should be the real semi-axis (0, ∞) because p/p * may be any positive number. Therefore, lim y→a g(y) = 0 and for finite a the function g is defined on [a, b). For each monotonically increasing function g on a real interval (a, b) with im g = (0, ∞), the corresponding solution of the Maxent problem is given by the distribution (3), where λ i are the solutions of the corresponding equation. This solution of the Maxent problem is the conditional minimizer of H h (P P * ) with h(x) = h ′ (x)dx, where h ′ (x) is the inverse function of g(y), i.e. h ′ (x) = y, where y is the solution to the equation g(y) = x. The additive constant in h ′ (x)dx does not affect the solution of any Maxent problems and may be chosen arbitrarily. Thus, we present the parametric description of the minimizers of all strictly convex divergences H h (P P * ). A monotonic function g with the values range (0, ∞) serves as a parameter in this description.
For the existence of a positive distribution P which satisfies (3) the moment conditions j m rj p j = M r (ρ = 0, . . . , k) should be compatible with the positivity of p i . Of course, for arbitrary g this may be not sufficient for the existence of such a positive distribution. To guarantee the existence of a positive Maxent distribution it is sufficient to add to the function h(x) a term εx ln x with arbitrarily small positive ε. This term creates a logarithmic singularity of h ′ (x) at zero. It is easy to check that this singularity guarantees the existence of a positive solution of (3) if the moment conditions are compatible with the positivity of p i . For some applied purposes an additional term −ε ln x may be even more convenient [17] because it guarantees the logarithmic singularity of entropy and h ′ (x) has the singularity ∼ −1/x at zero.
In this paper, the question about existence of the positive Maxent distribution is not important. We need only the conditions (3) which are necessary and sufficient for a positive distribution P = (p i ) to provide a minimizer of the given f -divergency under moment conditions.

Maxallent, approach #2: the Markov order
Any Markov process with equilibrium P * increases disorder. The classical Boltzmann-Gibbs-Shannon entropy grows in Markov processes. This theorem (the "data processing lemma") was proved in the first paper of Shannon [5] but of course the entropy growth in kinetics was known before (Boltzmann's H-theorem [2] and its generalization for the systems without detailed balance [18]).
A. Rényi proved in the first paper about the non-classical entropies [13] that all f -divergencies (1) decrease in Markov processes with equilibrium P * . Later on, it was demonstrated that this property characterizes f -divergencies among all functions which can be presented in the form of the sums over states (the "trace form") [21,22,23].
The generalized data processing lemma was proven [24,25]: For every two positive probability distributions P, Q the divergence H h (P Q) decreases under action of a stochastic matrix A = (a ij ) A second method of handling the UoU is based on a simple remark: "uncertainty of a probability distribution should increase in Markov processes". More precisely, let the most uncertain distribution P * be given (the equilibrium). If a distribution P ′ can be obtained from a distribution P in a Markov process with equilibrium P * then we can assume: uncertainty of P ≤ uncertainty of P ′ .
Thus, we do not care about the values of the uncertainty measure, we just compare the uncertainty of distributions: P ′ is more uncertain than P under given equilibrium P * (in this sense, the values vanish but the (pre)order appears [23]).
In the Maxent approach, the entropy is used as a (pre)order in the distribution space, not as a function, and the values are not important because any monotonically increasing transformation of the entropy does not change the solution of the Maxent problem. Of course, in some other applications the values of entropy are important: in coding theory (bits per symbol) and in thermodynamics (dU = T dS) the values of the entropy have a specific important sense. Nevertheless, when we discuss the entropy as a measure of uncertainty and work with the huge population of non-classical entropies, these entropies are, in their essence, (pre)orders on the space of distributions.
We consider the continuous time Markov processes with a given equilibrium distribution P * . By definition, the equilibrium is the unconditionally maximally uncertain distribution. To add the moment conditions we define a linear manifold in the space of distributions. For every non-equilibrium distribution P each Markov process with the equilibrium distribution P * determines the direction of P evolution, dP/dt. In this direction, the distribution becomes more uncertain. Let us take this property as a definition of the uncertainty. Instead of an entropy functional we use the transitive closure of this relation, define an order on the space of distributions and call it the "Markov order" [23].
Let Q(P, P * ) be a cone of possible time derivatives dP/dt for a given probability distribution P , the equilibrium P * , and all Markov processes with equilibrium P * .
For fixed values of moments, M r , the conditionally linear manifold L in the space of the probability distributions is given by equations j m rj p j = M r (r = 0, . . . , k). We can consider P 0 ∈ L as a possibly extremely disordered distribution on L, if for any Markov process with equilibrium P * the solution P (t) of the Kolmogorov equation with initial condition P (0) = P 0 has no points on the conditionally linear manifold L for t > 0 (we assume that P 0 is not a steady state for this process). Instead of this global condition, we consider the local condition (Fig. 1).
is a local minimum of the Markov order on L ∩ ∆ n−1 Further, for short, we can omit ∆ n−1 + and call P 0 "a local minimum of the Markov order on L". In this definition, we substitute the trajectories P (t) by their tangent directions at point P 0 , dP (t)/dt ∈ Q(P 0 , P * ). In Sec. 2 we justify this substitution and prove that the local condition (4) holds if and only if for every b) a) Figure 1: The local condition (4): P 0 may be an extremely disordered distribution on the condition linear manifold L if the set P 0 + Q(P 0 , P * ) intersects the linear manifold of conditions at the only point P 0 ; (a)(P 0 + Q(P, P * )) ∩ L = {P 0 } and P 0 may be an extremely disordered distribution on L; (b) (P 0 + Q(P, P * )) ∩ L {P 0 } and P 0 has no chance to be an extremely disordered distribution on L. In case (b), there are more disordered distributions on L achievable by the Markov processes from the initial distribution P 0 .
Markov process with equilibrium P * the solution P (t) of the Kolmogorov equation with initial condition P (0) = P 0 has no points on the condition linear manifold L for t > 0 (if P 0 is not a steady state for the process).
For applications, we need the local minima condition formalized by Definition 1 and the local order generated by the cone Q(P 0 , P * )) only. The general notion of (global) Markov order appears later, in Section 3, where we prove equivalence of the Maxima of all entropies and the Markov order approaches. Surprisingly, the set of the conditional minimizers of all f -divergencies and the set of the conditionally minimal elements of the Markov order coincide for the same conditions (Sec. 3). These sets include all reasonable hypotheses about conditionally most uncertain distributions. Let us call the problem of description of all the conditional minima of the Markov order the Maxallent problem.

Main tool: decomposition theorems
The main tools for constructive work with the Markov orders are the decomposition theorems for Markov chains. The first decomposition theorem states that every Markov chain with a positive equilibrium distribution is a convex combination of the simple directed cyclic Markov chains with the same equilibrium. The coefficients in this decomposition do not depend on the current probability distribution: the vector field dP/dt for a general Markov chain is a convex combination of these vector fields for simple cyclic Markov chains with the same positive equilibrium.
The second decomposition theorem states that for every Markov chain with a positive equilibrium distribution and for any non-equilibrium distribution P the velocity vector dP/dt is a convex combination of the velocity vectors for the simple cyclic Markov chains of the length two with the same equilibrium (i.e. of the reversible transitions between two states, A i ⇋ A j ). The coefficients in this decomposition typically depend on the current probability distribution.
The idea of the first decomposition theorem was used by Boltzmann in 1882 [18] in his proof of the H-theorem for systems without detailed balance. (This was his answer to the Lorentz objections [19].) He did not formulate this theorem separately but efficiently used the cycle decomposition for generalization of detailed balance. Later on, his extension of the detailed balance conditions were analyzed by many authors under different names as "cyclic balance", "semi-detailed balance" or "complex balance" (see, for example, the review [20]). Now, the theory of the cycle decomposition is a well developed area of the theory and applications of the random processes [26].
The second decomposition theorem is less known. We found this theorem in the analysis of the Markov order [23]. This decomposition means that for the general first-order kinetics and an arbitrary non-equilibrium probability distribution P there exists a system with detailed balance and the same equilibrium that has the same velocity dP/dt at point P [27]: the classes of the general Markov processes and the Markov processes with detailed balance are pointwise equivalent.
The decomposition theorems are discussed in Appendix B in more detail.

Local minima of Markov order
Let us consider continuous time Markov chains with n states A 1 , . . . , A n . The Kolmogorov equation (or master equation) for the probability distribution P = (p i ) is where q ij (i, j = 1, . . . , n, i = j) are non-negative. In this notation, q ij is the rate constant for the transition A j → A i . Any non-negative values of the coefficients q ij (i = j) correspond to a master equation. Therefore, the set of all the Kolmogorov equations (5) may be considered as the positive orthant R n(n−1) + in R n(n−1) with coordinates q ij (i = j). Now, let us restrict our consideration to the set of the Markov chains with the given positive equilibrium distribution P * (p * i > 0).
This system of uniform linear equations define a cone of the q ij (i, j = 1, . . . , n, i = j) in R n(n−1) + . Under the balance condition (6), the Kolmogorov equations (5) may be rewritten in a convenient equivalent form: We use below one of the f -divergencies (1) It is a quadratic divergence, the weighted l 2 distance between P and P * : With the master equation in the form (7), it is straightforward to calculate the time derivative of H 2 (P P * ) Each term in the sum is non-negative. The time derivative (8) is strictly negative if for a transition A j → A i the rate constant is positive, q ij > 0, and pi Hence, if the state P is not an equilibrium (i.e., the right hand side in (7) is not zero) then dH2(P P * ) dt < 0. An important class of the Markov chains is formed by reversible chains with detailed balance. The detailed balance condition reads: Under this condition, there are only n(n−1) 2 independent coefficients among n(n − 1) numbers q ij . For example, we can arbitrarily select q ij ≥ 0 for i > j and then take q ij = q ji p * i p * j for i < j. So, for given P * , the cone of the detailed balance systems (9) is a positive orthant in R are the convenient coordinates in R n(n−1) 2 for a description of the systems with detailed balance.
Let Q(P, P * ) be the set of all possible velocities dP/dt at a non-equilibrium distribution P for all Markov chains which obey a given positive equilibrium P * . According to the second decomposition theorem, the set of all possible velocities dP/dt for the chains with detailed balance and the same equilibrium is the same cone Q(P, P * ). Therefore, Q(P, P * ) is a convex polyhedral cone and its extreme rays consist of the velocity vectors for two-state Markov chains A i ⇋ A j with rate constants q ji = κ/p * j , q ji = κ/p * i (κ > 0). The construction of the cones of possible velocities was proposed in 1979 [28] for systems with detailed balance in the general setting, for nonlinear chemical kinetics. These systems are represented by stoichiometric equations of the elementary reactions coupled with the reverse reactions: where α ρi , β ρi ≥ 0 are the stoichiometric coefficient, ρ is the reaction number (ρ = 1, . . . , m). The stoichiometric vector of the ρth reaction is an n dimensional vector γ ρ with coordinates where w + ρ is the rate of the direct elementary reaction and w − ρ is the rate of the reverse reaction The equilibria of the ρth pair of reactions (10) form a hypersurface in the space of concentrations. The intersection of these surfaces for all ρ is the equilibrium (with detailed balance). Each surface of the equilibria of a pair of elementary reactions (10) divides the non-negative orthant of concentrations into three sets: (i) w ρ > 0, (ii) w ρ = 0 (the surface of the equilibria) and (iii) w ρ < 0. All the surfaces of equilibria (w ρ = 0) divide the non-negative orthant of concentrations into compartments. In each compartment, the dominant direction of each reaction (10) is fixed and, hence, the cone of possible velocities is also constant. It is a piecewise constant function of concentrations: where "cone" stands for the conic hull, that is the set of all linear combinations with non-negative coefficients.
Here and below we use the three-valued sign function (with values ±1 and 0).
Let us apply this construction to Markov chains with detailed balance. Let us join the transitions A i ⇋ A j in pairs (say, i > j) and introduce the stoichiometric vectors γ ji with coordinates: Let us rewrite the Kolmogorov equation for the Markov process with detailed balance (9) in the quasichemical form: Here, w * ij = q ij p * j = q ji p * i is the equilibrium flux from A i to A j and back. The cone of possible velocities for (12) is The standard simplex of distributions P is divided by linear manifolds pi are the polyhedra where the cone of the local Markov order Q(P, P * ) is constant. The compartments for the Markov chains with the positive equilibrium P * correspond to various partial orders on the finite set {p i /p * i } (i = 1, . . . , n). Let us describe the compartments and cones in more detail following [23]. For every natural number k ≤ n − 1 the k-dimensional compartments are enumerated by surjective functions σ : {1, 2, . . . , n} → {1, 2, . . . , k + 1}. Such a function defines the partial ordering of quantities pj p * j inside the compartment: Let C σ be the corresponding compartment and Q σ be the corresponding local Markov order cone (Q(P, For a given surjection σ the compartment C σ and the cone Q σ have the following description: In Fig. 2, the partition of the standard distribution simplex into compartments, and the cones (angles) of possible velocities are presented for the Markov chains with three states. In the construction of this cone, reversible chains with detailed balance are used. Due to the second decomposition theorem, this construction of the cone of possible velocities is valid for the class of general Markov chains (and not only for reversible chains) with the same equilibrium. It seems quite surprising that the Markov order for general Markov chains is generated by the reversible Markov chains which satisfy the detailed balance principle.
Let L be a linear manifold in the probability distribution space. Due to Definition 1, P 0 ∈ L ∩ ∆ n−1 + is a local minimum of the Markov order on L ∩ ∆ n−1 + if the condition (4) holds.  (Fig. 3a,b). If L is parallel to one side of the triangle (Fig. 3c) then the moments are just some of the p i and the extreme points of the Markov order on L ∩ ∆ n−1 + coincide with the partial equilibria.
Let J be a set of pairs of indexes (i, j) (i > j) and K J be the class of kinetic equations (12) with w * ij = 0 for (i, j) / ∈ J and w * ij ≥ 0 for (i, j) ∈ J (i = j). We define Φ J (P 0 ) for an initial distribution P 0 as a set of all values P (t) (t > 0) for solutions P (t) of all equations from the class K J with initial value P (0) = P 0 .
Consider a cone of possible velocities for the set of transitions A i ⇋ A j , (i, j) ∈ J: The following proposition states that in a vicinity of the distribution P 0 the sets Φ J (P 0 ) and P 0 + Q J (P 0 , P * ) coincide. This gives a justification of the use of the cone of the tangent directions Q J (P 0 , P * ) in the definition of the local minima of the Markov order (4).
for a distribution P = P 0 . There exists a vicinity U of P 0 where P 0 + Q J (P 0 , P * ) coincides with Φ J (P 0 ): Proof. There exists a Euclidean ball B r around P 0 where pj p * j − pi p * i = 0 ((i, j) ∈ J). Due to (8), inside B r , the divergence H 2 (P P * ) strictly decreases with λ increasing along any ray P 0 + λe, e ∈ Q J (P, P * ) (λ > 0). For each ray, we can find the minimum of H 2 (P P * ) in B r . Let the maximum of these minima be h r (P 0 ): By construction, H 2 (P 0 P * ) > h r (P 0 ). The set is a vicinity of P 0 . The intersection (P 0 + Q J (P 0 , P * )) ∩ U is For any system from K J on B r and for any distribution P ∈ (P 0 + Q J (P 0 , P * )) the velocity vector dP/dt belongs to Q J (P 0 , P * ). Obviously, (P + Q J (P 0 , P * )) ⊂ (P 0 + Q J (P 0 , P * )). Therefore, the solution of this system with the initial condition P (0) = P 0 may leave the intersection (P 0 + Q J (P 0 , P * )) ∩ U through the level surface H 2 (P 0 P * ) = h r (P 0 ) only. After that, the solution cannot return to U because in U the value of H 2 (P 0 P * ) are bigger, H 2 (P 0 P * ) > h r (P 0 ), and H 2 (P (t) P * ) should decrease in time along every solution of any system from K J . Thus, one inclusion is proven, To prove the second inclusion, (P 0 + Q J (P 0 , P * )) ∩ U ⊆ Φ J (P 0 ) ∩ U , we have to demonstrate that the solutions P (t) (P (0) = P 0 , t ≥ 0) of the equations from K J cover (P 0 + Q J (P 0 , P * )) in some vicinity of P 0 .
The polyhedral cone Q J (P 0 , P * ) is covered by the simplicial cones spanned by the sets of linearly independent vectors γ ji sign( ). Therefore, it is sufficient to prove the second inclusion for the simplicial cones Q J (P 0 , P * ). Let the vectors {γ ji |(i, j) ∈ J} be linearly independent. For the simplicity of notation, let us enumerate the states in the order of the values of p 0 j /p * j : In these notations, sign( Consider a subset of the cone Q J (P 0 , P * ) (a "pyramid") The "base" of this pyramid is a simplex Let α > 0 be sufficiently small and, therefore, in P 0 + αQ J (P 0 , P * ). For this α, a solution P (t) (t ≥ 0) of an equation from the class K J with initial data P (0) = P 0 may leave P 0 + αQ J (P 0 , P * ) only through its base, P 0 + αB J (P, P * ). Let us prove that if α is sufficiently small then for each point y ∈ B J (P, P * ) there exists a system in K J whose solution P (t) (t > 0, P (0) = P 0 ) leaves P 0 + aQ J (P 0 , P * ) through the point P 0 + αy. This means that P (t 1 ) = P 0 + αx for some t 1 > 0 and P (t) ∈ Q J (P, P * ) for 0 < t < t 1 .
Each vector x ∈ B J (P, P * ) can be expanded into a linear combination of γ ji ((i, j) ∈ J): With this expansion we define the system K x ∈ K J by the condition dP dt P =P 0 = x: (12).) A solution P (t) (P (0) = P 0 ) of this equation (18) can be also expanded into a linear combination of γ ji ((i, j) ∈ J) (x ∈ B J (P, P * )): where ν ij (t, {θ lm }) are analytic functions. If x belongs to a face F of the cone Q J (P 0 , P * ) then P (t) ∈ P 0 +F for sufficiently small t. The moment t = t(α, x) when the solution P (t) (19) leaves P 0 + αQ J (P 0 , P * ) is a root of equation Due to the standard inverse function theorems this root exists and the function t(α, x) is smooth for sufficiently small α for all x ∈ B J (P, P * ), and t(α, x) = α + o(α). The solution P (t) (19) of the system K x (18) leaves P 0 + αQ J (P 0 , P * ) at the point P (t(α, x)) = P 0 + αy(x), where y(x) ∈ B J (P, P * ).
To prove that y(•) : B J (P, P * ) → B J (P, P * ) is a homeomorphism of the simplex B J (P, P * ) onto itself, let us notice that the map x → y(x) leaves the faces of the simplex B J (P, P * ) invariant: vertices transform into themselves, the same for edges, etc.
We use the following topological lemma, the multidimensional intermediate value theorem. Consider a continuous map Ψ : ∆ n → ∆ n of the n-dimensional standard simplex into itself. Let each face F ⊂ ∆ n be Ψ-invariant, i.e. Ψ(F ) ⊂ F . Then Ψ is surjective. The proof is possible by induction in n: for n = 0 it is obvious, for n = 1 this is just a 1D intermediate value theorem. In all dimensions, it can be proved on the basis of the "no-retraction theorem" [29] and simple inductive topological reasoning, which reduces the general case to the situation when all the faces F ⊂ ∆ n consist of fixed points of the map Ψ.
Therefore, for sufficiently small α the solutions P (t) (P (0) = P 0 , t ≥ 0) of the equations from K J cover (P 0 + aQ J (P 0 , P * )) in some vicinity of P 0 . The second inclusion is proven. Let us combine the inclusions and reduce the vicinities, if necessary.
for P in some vicinity of P 0 . If for some pairs i, j (i = j) Fig. 3c) then for some P ∈ P 0 +Q(P 0 , P * ) the cone Q(P, P * ) may be bigger than Q(P 0 , P * ) even in a small vicinity of P 0 . Nevertheless, the set of trajectories P (t) (t > 0, P (0) = P 0 ) remains in P 0 + Q(P 0 , P * ) for sufficiently small t. Let us prove this statement.
Let K be the class of all master equations with detailed balance with the positive equilibrium P * (12) with w * ij ≥ 0 for all (i, j) (i > j). We define Φ(P 0 ) for an initial distribution P 0 as a set of all values P (t) (t > 0) for solutions P (t) of all equations from the class K with initial value P (0) = P 0 . Proposition 2. For every probability distribution P 0 there exists a vicinity U of P 0 where P 0 + Q(P 0 , P * ) coincides with Φ(P 0 ): (P 0 + Q(P 0 , P * )) ∩ U = Φ(P 0 ) ∩ U .
Proof. The inclusion (P 0 + Q(P 0 , P * )) ∩ U ⊂ Φ(P 0 ) ∩ U is proven in the second part of the proof of Proposition 1 because K J ⊂ K. We have to prove the inclusion (P 0 + Q(P 0 , P * )) ∩ U ⊃ Φ(P 0 ) ∩ U . Let us use the combinatorial description of compartments and cones (15). We assume that P 0 ∈ C σ for a surjection σ : {1, 2, . . . , n} → {1, 2, . . . , k + 1}. Let us recall that k = dim C σ . If k = n − 1 then C σ is an open subset of the distribution space and the preimage of every l = 1, 2, . . . , n consists of one point. For every P ∈ C σ the cone Q(P, P * ) coincides with Q(P 0 , P * ) and due to Proposition 1 there exists a vicinity U of P 0 where P 0 + Q(P 0 , P * ) coincides with Φ(P 0 ).
Let us prove that for any Markov chain with equilibrium P * for sufficiently small time τ > 0 and a ball U r/2 with the centre P 0 the solutions of the Kolmogorov equations P (t) do not leave P 0 + Q(P 0 , P * ) during the time interval [0, τ ] if P (0) ∈ (P 0 + Q(P 0 , P * )) ∩ U r/2 .
A set V is positively invariant with respect to a dynamical system if every motion that starts in V at t = 0 remains there for t > 0. Let a convex set V be positively invariant with respect to several dynamical system given by Lipschitz vector fields w 1 , . . . , w r . Then V is positively invariant with respect to any combination w = j f j w j , where f j are non-negative functions and w is a Lipschitz vector field. Therefore, the problem of positive invariance of a convex set with respect to such combinations of vector fields can be "split" into problems of the positive invariance of V with respect to summands w j . Due to the second decomposition theorem, we can always assume that the vector field of the Kolmogorov equation for the Markov kinetics is a linear combination of the vector fields of the pairs of elementary transitions A i ⇋ A j with the same equilibrium. The coefficients in these combinations are non-negative functions.
The motion P (t) with P (0) ∈ (P 0 + Q(P 0 , P * )) does not leave The cone Q(P 0 , P * ) is generated by vectors γ ij with σ(j) = σ(i) + 1. To generate a cone Q(P, P * ) for a point P ∈ U r we have to add to the set of γ ij (σ(j) = σ(i) + 1) some of γ ij with σ(j) = σ(i). Let us consider the pyramid (compare to (16)) We will prove that the set P 0 + aQ(P 0 ) is positively invariant with respect to any first order kinetics with transitions A i ⇋ A j (i, j ∈ S l ) and equilibrium P * for any l = 1, . . . , k + 1.
It is sufficient to consider dynamics in projections on the coordinate subspace R S l with coordinates p i , i ∈ S l for every l ∈ I separately. In this space, vectors γ ij (i, j ∈ S l ) correspond to the standard first order kinetics like (12) with the reduced vector P ∈ R S l but without compulsory unit balance ( i∈S l p i = const with any const > 0). A projection of P 0 on R S l , P 0 S l is an equilibrium for this first order kinetics with the balance i∈S l p i = i∈S l p 0 i because The vectors γ ij that generate Q(P 0 , P * ) (σ(j) = σ(i) + 1) (20) have non-zero projections on R S l if and only if either l = σ(j) = σ(i) + 1 or σ(j) = σ(i) + 1 = l + 1. In the first case, l = σ(j) = σ(i) + 1, vector γ ij is the standard basis vector e j in R S l . In the second case, σ(j) = σ(i) + 1 = l + 1, we have γ ij = −e i . If l = 1 then only the second case is possible, and if l = k + 1 then only the first case can take place.
Let V l = {P ∈ R S l | p i ≥ 0, i∈S l p i < 1}. The projection of the pyramid Q(P 0 ) onto R S l is conv(V l − V l )) if 1 < l < k + 1; it is V l if l = k + 1 and −V l if l = 1. (For sets X, Y , the sum X + Y is the set of all sums x + y (x ∈ X, y ∈ Y ), the difference X − Y is the set of all differences x − y, therefore V − V is not {0} if V includes more than one element.) The set V l is positively invariant with respect to the first order kinetics in R S l . Therefore, the following sets are also positively invariant with respect to the first order kinetics in R S l with equilibrium P 0 S l for every a > 0: P 0 S l + aV l , (P 0 S l + aV l ) − aV l , conv((P 0 S l + aV l ) − aV l ) . Thus, the set P 0 + aQ(P 0 ) is positively invariant with respect to any first order kinetics with transitions A i ⇋ A j (i, j ∈ S l ) and equilibrium P * for any l = 1, . . . , k + 1. A combination of these statements for all l = 1, . . . , k + 1 finalizes the proof.
This proposition finalizes the justification of the use of the cone of the tangent directions Q(P 0 , P * ) in the definition of the local minimum of the Markov order (4).

Equivalence of the maxima of all entropies and the Markov order approaches
The cone Q(P 0 , P * ) is a piecewise constant function of P 0 : it is the same for all P 0 from one compartment C σ and, hence, depends on σ only. Therefore, if the condition of the local minimum (4) holds for one P 0 ∈ L ∩ C σ then it holds also for all elements of L ∩ C σ . There is a finite number of compartments C σ .
Let the linear manifold of conditions L be given by the values of moments i m ri p i = M r , L 0 = ker m and L ∩ ∆ n−1 + = ∅. The set of all conditional local minima of the Markov order on the linear manifold of conditions L is where C σ and Q σ are defined by (15). It is sufficient to find all σ such that L ∩ C σ = ∅ and L 0 ∩ Q σ = {0} and then describe the union of the compartments C σ for these σ.
The approach based on the minimization of all f -divergencies seems to be very different. For all monotonically increasing functions g we have to solve the equations for the Lagrange multipliers and represent the probability distribution in the form (3). Nevertheless, these approaches are equivalent and describe the same set of the "conditionally maximally disordered distributions". Proof. Due to the classical theorems about separation of convex sets and linear spaces by linear functionals [30], a distribution P 0 satisfies the condition of the local minimum (4) if and only if there exists a linear functional ψ(P ) = i ψ i p i such that ψ| L = ψ(P 0 ) = const and ψ(P ) > ψ(P 0 ) for every P ∈ P 0 +Q(P 0 , P * ) if P = P 0 . In other words, ψ| L 0 ≡ 0 and according to the definition of Q(P, P * ) (13). Condition ψ| L 0 ≡ 0 is equivalent to the existence of the coefficients λ r such that for all i ψ i = r λ r m ri .
Condition (22) is equivalent to the existence of a strictly monotonic function η(x) defined for x ≥ 0 such that To find such a function η(x) we can take the known values ψ i for x = p 0 i /p * i and then use, for example, linear interpolation η(x) between p 0 i /p * i . To extrapolate η(x) from max{p 0 i /p * i } to +∞ we can use an increasing linear function. To extrapolate η(x) on the interval (0, min{p 0 i /p * i }) we can use ε log x + const. Finally, we can take h ′ (x) = η(x), h(x) = η(ξ) dξ; and g(y) is the inverse function: g(η(x)) = x for x ≥ 0. The distribution P 0 is the local minimum of H h (P P * ) on L.
Conversely, if P 0 is a minimum of a strictly convex Lyapunov function H on L and dH/dt| P 0 < 0 for every Markov chain with equilibrium P * for which P 0 is a non-equilibrium distribution then we can take This choice of ψ i provides (22) (because H is strictly decreasing in time Lyapunov function) and ψ| L 0 ≡ 0 because gradH is orthogonal to L (the condition of local minimum).
This equivalence of two definitions of the maximally uncertain distribution under given conditions has several important consequences.
Let us introduce the notion of the (global) Markov order [23].
• If for distributions P 0 and P 1 there exists such a Markov process with equilibrium P * that for the solution of the Kolmogorov equation with P (0) = P 0 we have P (1) = P 1 then we say that P 0 and P 1 are connected by the Markov preorder [23] with equilibrium P * and use notation P 0 ≻ 0 P * P 1 . • The (global) Markov order is the closed transitive closure of the Markov preorder. For the Markov order with equilibrium P * we use notation P 0 ≻ P * P 1 .
The local Markov order at point P 0 is just a vector order generated by the tangent cone Q(P 0 , P * ) [23]. We use for this local order the notation > P 0 ,P * : The proofs of Propositions 1 and 2 give us the possibility to use the relation P 0 > P 0 ,P * P 1 instead of the Markov preorder for the definition of the Markov order minimizers on linear manifolds. The relation P 0 > P 0 ,P * P 1 is defined by the local Markov order in a vicinity of P 0 : cannot have an opposite order on the real line. The closed transitive closure of the relation P 0 > 0 P * P 1 is the Markov order P 0 ≻ P * P 1 Let L be a linear manifold in the space of distributions. By definition, P 0 ∈ L is a minimal point on L ∩ ∆ n−1 + with respect to the order ≻ P * if and only if there is no point P 1 ∈ L ∩ ∆ n−1 + , P 1 = P 0 such that P 0 ≻ P * P 1 . Proof. If P 0 ∈ L ∩ ∆ n−1 + is a minimal point on L ∩ ∆ n−1 + with respect to the (global) Markov order then it satisfies the condition (4) due to the definition of the Markov order through the transitive closure of the relation P 0 ≻ 0 P * P 1 and Propositions 1 and 2. Let P 0 satisfy the local minimum condition (4). Then there exists a divergence H h (P P * ) with strictly convex h(x) (x ≥ 0) such that P 0 is a local minimum of H h (P P * ) on L. Because of strong convexity, this local minimum is a global one. H h (P P * ) is a Lyapunov function for all Markov chains with equilibrium P * . Therefore, a broken line, which is combined from solutions of the Kolmogorov equations for such Markov chains and starts at P 0 , leaves a small vicinity of P 0 (Propositions 1 and 2) and never returns in a sufficiently small vicinity of L. Thus, for the closed transitive closure of the relation P ≻ 0 P * P ′ , point P 0 is a minimal point on L.
Of course, there may be infinitely many minimal points of the Markov order on L and each of them corresponds to a different Lyapunov functions H h (P P * ).
Another remarkable order on the space of distributions is P 0 > H,P * P 1 if for all strictly convex functions that is, P 1 is closer to equilibrium than P 0 with respect to all divergencies H h (P P * ). Corollary 2. For any linear manifold L in the distribution space the minimal elements of the Markov order ≻ P * on L ∩ ∆ n−1 + coincide with the minimal elements of the order > H,P * on L ∩ ∆ n−1 + . Proof. We just have to combine Theorem 1 with Corollary 1.
Thus, the minimal elements of the orders ≻ P * and > H,P * on the linear manifolds coincide. Nevertheless, it is necessary to mention the difference between these orders. Let P 0 be a distribution. For > H,P * the set of distributions {P | P 0 > H,P * P } is convex as an intersection of convex sets {P | H h (P 0 P * ) > H h (P P * )} for various strictly convex h. This is not the case for the Markov order. The set of distributions {P | P 0 ≻ P * P } may be non-convex. The examples may be extracted from the papers [28,31] (see Fig. 4).

Example: generalization of the normal distribution
In this section, we discuss distributions p(x) on a continuous space of states, the non-negative real semiaxis, R + = {x | x ≥ 0}. We have in mind two classical examples of distributions of the quantity bounded from below: energy (physics) and wealth (economics and microeconomics).
Let two moments be fixed, the total probability M 0 = ∞ 0 p(x) dx and the average quantity M 1 = ∞ 0 xp(x) dx. The conditional maximization of the classical Boltzmann-Gibbs-Shannon entropy gives: This Boltzmann distribution appears always as a first candidate for the equilibrium distribution of an additive conserved quantity bounded from below. Khinchin (1943) clearly explained this law as a version of the limit theorem [32].
Technically, it is not difficult to involve the higher moments and obtain the distribution of the form One can expect that this extension of the set of moments may improve the description. This is a traditional belief in Extended Irreversible Thermodynamics (EIT) [7]. There may be many different approaches to evaluation of the quality of the approximation (24) but at least one important property of these functions is wrong: the asymptotic behavior at large x is p(x) ≍ exp(−const × x r ). These "super-light" tails of the distribution p(x) change qualitatively with the change of the order r in (24).
If we use, for example, the "regularizing" forth moment in the moment chain for the Boltzmann equation [33] then we corrupt the e −const×v 2 tails of the Maxwell distribution. Therefore, other approaches which do not modify the tails of the distribution qualitatively (like [8]) may be more appreciated.
The asymptotic behavior of the distribution's tails was thoroughly studied in many cases. Very often, the tails of the distributions are, without a doubt, heavier than normal e −const×x 2 and definitely are not cut as e −const×x 4 . For example, it is demonstrated that the distribution of money between people has the exponential tail with a possible transformation into a heavier power tail for very rich people [34].
The general solution (3) with the Boltzmann equilibrium (23) gives the following expression instead of (24) where g is a monotonically increasing function. In particular, for the moments M 0 , M 1 and M 2 we obtain There are four qualitatively different cases of (25). Let λ 2 = 0 and µ = − λ1 2λ2 . Then Each of these "generalized normal distributions" (26) is a minimizer of the corresponding f -divergence. For the construction of such a divergence in general case, it is convenient to define the convex functions h in (1) with values on an extended real line with additional possible value +∞. This is a natural general definition of convex functions [30]. In case 1 (µ ≤ 0, λ 2 > 0, and f increases), we can take in (25), (26) without loss of generality µ = 0, f (x) = g(x 2 ), and g(y) = f ( √ y). The monotonically increasing function g(y) is, therefore, defined on [0, ∞) with the set of values [g, g), where g = f (0) ≥ 0, g = lim x→∞ f (x) > 0 and the upper limit may be finite or infinite. The inverse function ξ(z) is defined for z ∈ [g, g) with the interval of values [0, ∞). Let us take The improper integral g g ξ(ς)dς may take finite or infinite values.
Similarly, in case 2 (µ ≤ 0 and λ 2 < 0, f decreases) we define g(y) = f ( √ −y) for y ∈ (−∞, 0]. The function g(y) monotonically increases and takes values on (g, g], where g = lim x→∞ f (x) and g = f (0). The inverse function ξ(z) is defined for z ∈ (g, g] with the interval of values (−∞, 0]. In this case, we can take In case 3, the construction is almost the same as for the case 1 but f (x) = g((x − µ) 2 ) and g(y) = f ( √ y + µ). In this case, g(y) is a monotonically increasing function defined on the interval [0, ∞) with the set of values [g, g), where g = f (µ) and g = lim x→∞ f (x). Similarly, for case 4, the construction of h(z) is almost the same as in case 2.
Thus, for every distribution in the form (26) we can find a f -divergence H h (P P * ), which conditional minimization produces this distribution. For example, if in (26) f (x) = ax β then we can take h in the form h(z) = β β+2 (z/a) 1+2/β .

Conclusion
The Maxallent approach aims to bring some order to the modern anarchy of the measures of disorder. If there is no clear idea which entropy is better then we have to use all of them together.
The Markov order approach was also proposed as an alternative to the entropic anarchism. It is based on the idea that the disorder has to increase in random processes with given equilibrium distribution, which is considered as the maximally disordered state. Here, we have proved that these two approaches produce the same conditional minimizers on the planes of given values of moments (Theorem 1).
In this paper, we have considered several relations between positive distributions: 1. P 0 ≻ 0 P * P 1 if there exists a Markov chain with equilibrium P * such that for the solution of the Kolmogorov equation P (t) with P (0) = P 0 we have P (1) = P 1 ; 2. P 0 ≻ P * P 1 if there exist integrable bounded functions q ij (t) (i, j = 1, . . . , n, i = j, t ≥ 0) such that q ij (t) satisfy the balance condition (6) for given P * (p * i > 0) (for all t ≥ 0), and P (1) = P 1 for solution P (t) of the equations dp i dt = j, j =i (q ij (t)p j − q ji (t)p i ) (i = 1, . . . , n) with P (0) = P 0 (that is, ≻ P * is the transitive closure of ≻ 0 P * ); 3. P 0 > H,P * P 1 if H h (P 0 P * ) > H h (P 1 P * ) for all strictly convex functions h(x) on a semi-axis x ≥ 0. 4. P 0 > P 0 ,P * P 1 if P 1 − P 0 ∈ Q(P 0 , P * ), where Q(P 0 , P * ) is the cone of possible velocities dP/dt (13) at point P 0 for all Markov chains with equilibrium P * .
All these relations are different. Three of them are antisymmetric, and one, P 0 > P 0 ,P * P 1 , is locally antisymmetric, in a vicinity of P 0 . Their interrelations are described by the follows implications: The local Markov order P 0 > P * P 1 is the weakest and the connection by a solution of the Kolmogorov equation P 0 ≻ 0 P * P 1 is the strongest of these relations. Nevertheless, locally, in a small vicinity of a positive non-equilibrium distribution P 0 , these relations coincide and they define the same set of locally minimal distributions on a linear manifold of conditions L (Propositions 1, 2, Theorem 1, Corollaries 1, 2 and 3).
Of course, there is the other, the classical way to reduce the variability of the measures of disorder. The divergences H(P P * ) can be defined by their main properties. This is an axiomatic approach: we postulate some "natural properties" of the divergence, then find the divergences with these properties, evaluate the result and decide whether we have to change the system of axiom or not. The axiomatic approach to definition of entropy was used by Shannon [5] and elaborated in detail by Khinchin [35].
Two distinguished additivity properties are important for the Maxent reasoning: • Additivity on the algebra of states: H(P P * ) is a sum in states • Additivity with respect to the joining of independent subsystems. This means that if P and P * are products of distributions then H(P P * ) is the sum of the corresponding entropies: if P = (p jl ) = (q j r l ) and P * = (p * jl ) = (q * j r * l ) then H(P P * ) = H(Q Q * ) + H(R R * ). If we join the first additivity property with the requirement that the divergence should be a Lyapunov function for all Markov chains with equilibrium P * then we get H h (P P * ) of the form (1) [21,22,23]. If we add the second additivity property and require continuity of H h (P P * ) for all values of P (including vectors with some p i = 0) then the classical Boltzmann-Gibbs-Shannon relative entropy will be the only possibility (that is, H h (P P * ) with h(x) = x ln x up to unimportant constant factors and summand). If we relax the requirement of the continuity to the set of strictly positive distributions then we will get the one-parametric family H h (P P * ) with h(x) = βx ln x − (1 − β) ln x [21,23].
Let us accept the point of view that the divergency is an order. Then the values are not important and all the divergencies connected by a monotonic transformation of a scale, H = f (H ′ ) (with a monotonically increasing f ), are equivalent. If the first additivity property is valid in one scale, and the second may be valid in another one, then one more one-parametric family appear, the Cressie-Read divergences (see Appendix A) [21,23]. The Tsallis entropy is a particular case of them. The Boltzmann-Gibbs-Shannon relative entropy (or the Kullback-Leibler entropy, which is the same), the convex combination of H h (P P * ) and H h (P * P ) for h(x) = x ln x, and the Cressie-Read divergences (including the Tsallis relative entropy) form the "entropic aristocracy" distinguished mostly by the additivity properties.
If we accept the additivity on the algebra of states (i.e., the trace form) and the additivity with respect to joining of independent subsystems, both, then we have to use some of these functions. If additivity with respect to joining of independent subsystems seems to be too restrictive then we have to take the wider class of divergencies, for example, H h (P P * ) of the form (1). If we reject the requirement of the trace form then the variety of the admissible divergences becomes even richer. This uncertainty in the choice of divergence forces us to use the Maxallent approach.
The Maxallent approach produces a set of conditionally maximally disordered distributions instead of a single distribution that maximizes a selected distinguished entropy in the usual Maxent method. These Maxallent sets of distributions may be considered as probabilistic analogues of the type-2 fuzzy sets introduced by L. Zadeh [36] to capture the uncertainty of the fuzzy systems. The Maxallent approach is invented to manage the uncertainty of the measures of uncertainty. If there is no uncertainty of uncertainty then the set of distributions reduces to a single distribution.
The decomposition theorems for Markov chains provide us with tools for the efficient calculation of the Markov order. Following [27], we compare the general Markov chains and the reversible chains with detailed balance. For any general chain there is a reversible chain with the same velocity vector at a given point. The classes of general and reversible chains locally coincide because they have the same cone of possible velocities at every non-equilibrium distribution (the second decomposition theorem, Appendix B). This theorem gives us the possibility to describe the set of the conditionally maximally uncertain distributions combinatorially, in the finite form (21).
For the classical Boltzmann-Gibbs-Shannon entropy the distribution on R + with two given moments has the Gaussian form a exp(−b(x−c) 2 ). The class of the Maxallent distributions on R + with two given moments is also simple (26) but much richer. It can be produced by multiplication of the Boltzmann distribution (23) by a monotonic function or unimodal function (with one local maximum) or by a function with one local minimum.
There exists an attractive possibility: if a distribution can be obtained in the Maxallent approach then it is a conditional minimum of a divergence. If we find or guess a distribution of the Maxallent type for an empirical system then we can restore the divergence and then use it in the standard Maxent reasoning.
The Maxallent approach is, surprisingly, efficient enough to analyze some practical problems. It gives an answer that does not depend on the subjective choice and, therefore, returns us to the "mission" of information theory: "to eliminate the psychological factors involved..." [4]. At the same time, it has a solid basis in the theory of Lyapunov functions for the Kolmogorov equations. Now, essential mathematical work on the basic notion of entropy is needed. Gromov suggests that the natural mathematical language for this work will involve nonstandard analysis and category theory [37]. These abstract languages seem to be closer to the basic intuition than the set theory of Cantor and the ε − δ reasoning of the classical analysis. Nevertheless, the basic idea of Maxallent is so simple and natural, that it should persist in the future advanced theory of entropy: order is something that decreases in Markov processes. The inequality κ ≥ 0 holds because p 1 /p * 1 is the minimal value of the flux p j /p * j . We just delete the vertex with the smallest outgoing flux from the initial cycle of length n and add a cycle of the length 2 with the same equilibrium. Let us repeat this operation for the remaining cycle of the length n − 1, and so on. At the end, the left hand side vector v n will be represented as the combination with positive coefficients the vectors dP/dt for the cycles of the length 2, A i ⇋ A j with the same equilibrium. This is the system with detailed balance. We have to stress here that the set of these transitions and the coefficients κ depend on the current distribution P .
For every distribution P , the velocity dP/dt of every cycle with equilibrium P * is a combination with positive coefficients of the velocities for some cycles of the length two A i ⇋ A j with the same equilibrium. Therefore, the right hand side of the Kolmogorov equation for any Markov chain with equilibrium P * also allows such a decomposition.
It is necessary to stress that the decomposition of the right hand side of the Kolmogorov equation (5) into a conic combination of cycles of length 2 depends on the ordering of the ratios p i /p * i and cannot be performed for all values of P simultaneously.
For more details and further references see [27].