Statistical Inference on Random Structures

Generalized statistics framework for rate distortion theory

“…This forecloses the prospect of transparently establishing the symmetry of the generalized mutual entropy in the range 0 < q < 1, in the manner akin to (9), and, the Boltzmann-Gibbs-Shannon model.…”

“…Further, for normalized Tsallis entropies, Yamano [11] has elegantly established the symmetry described by (9) in the range 0 < q < 1.…”

“…Theorem 3 acquires a certain significance especially when it may be proven [11] that the convex form of the generalized mutual entropy (10) can never be expressed in the form of Tsallis entropies (9). Theorem 3 demonstrates that such a relation is indeed possible by commencing with q * -Tsallis mutual entropy and performing manipulations that scale the generalized mutual entropy from q * -space to q-space, yielding a form akin to (9). Interchanging the range of values and the connotations of q and q * respectively, such that 0 < q < 1 ( q * > 1), Theorem 3 may be modified to justify defining the convex q-Tsallis mutual entropy by (10).…”

“…Here, (a) follows from (5), (b) follows from setting q * = 2 − q, and, (c) follows from (9). Theorem 3 acquires a certain significance especially when it may be proven [11] that the convex form of the generalized mutual entropy (10) can never be expressed in the form of Tsallis entropies (9).…”

“…The representation of RD theory in the form of a variational principle, expressed within the framework of the Shannon information theory, has been established [3]. The computational implementation of the RD problem is achieved by application of the BlahutArimoto alternating minimization algorithm [3,8], derived from the celebrated Csiszár-Tusnády theory [9].…”