A Deformed Exponential Statistical Manifold

• Main Authors: Francisca Leidmar Josué Vieira, Luiza Helena Félix de Andrade, Rui Facundo Vigelis, Charles Casimiro Cavalcante
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Entropy
• Subjects: deformed exponential manifold; statistical manifold; <i>φ</i>-family; information geometry; exponential arcs
• Online Access: https://www.mdpi.com/1099-4300/21/5/496

Consider <inline-formula> <math display="inline"> <semantics> <mi>&#956;</mi> </semantics> </math> </inline-formula> a probability measure and <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">P</mi> <mi>&#956;</mi> </msub> </semantics> </math> </inline-formula> the set of <inline-formula> <math display="inline"> <semantics> <mi>&#956;</mi> </semantics> </math> </inline-formula>-equivalent strictly positive probability densities. To endow <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">P</mi> <mi>&#956;</mi> </msub> </semantics> </math> </inline-formula> with a structure of a <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>&#8734;</mo> </msup> </semantics> </math> </inline-formula>-Banach manifold we use the <inline-formula> <math display="inline"> <semantics> <mi>&#966;</mi> </semantics> </math> </inline-formula>-connection by an open arc, where <inline-formula> <math display="inline"> <semantics> <mi>&#966;</mi> </semantics> </math> </inline-formula> is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the <i>q</i>-deformed exponential and <inline-formula> <math display="inline"> <semantics> <mi>&#954;</mi> </semantics> </math> </inline-formula>-exponential functions. Moreover, we find the tangent space of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">P</mi> <mi>&#956;</mi> </msub> </semantics> </math> </inline-formula> at a point <i>p</i>, and as a consequence the tangent bundle of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">P</mi> <mi>&#956;</mi> </msub> </semantics> </math> </inline-formula>. We define a divergence using the <i>q</i>-exponential function and we prove that this divergence is related to the <i>q</i>-divergence already known from the literature. We also show that <i>q</i>-exponential and <inline-formula> <math display="inline"> <semantics> <mi>&#954;</mi> </semantics> </math> </inline-formula>-exponential functions can be used to generalize of R&#233;nyi divergence.