Information Geometry - Introduction - Alpha Connection

Alpha Connection

A non-metric connection is not determined by a metric tensor ; instead, it is and restricted by the requirement that the parallel transport between points and must be a linear combination of the base vectors in . Here,

expresses the parallel transport of as linear combination of the base vectors in, i.e. the new minus the change. Note that it is not a tensor (does not transform as a tensor).

For such a metric, one can construct a dual connection to make


for parallel transport using and .

For the mentioned -families the affine connection is called the -connection and can also be expressed in more ways.

For :

  • is a metric connection and with .
  • ,

    i.e. is dual to with respect to the Fisher metric.

  • If this is called -affine. Its dual is then -affine.

    , i.e. 0-affine, and hence, i.e. 1-affine.

Read more about this topic:  Information Geometry, Introduction

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