  Probability distributions can be read as simple expressions of information.
Each continuous probability distribution describes how information changes with
magnitude. Once one learns to read a probability distribution as a measurement
scale of information, opportunities arise to understand the processes that
generate the commonly observed patterns. Probability expressions may be parsed
into four components: the dissipation of all information, except the
preservation of average values, taken over the measurement scale that relates
changes in observed values to changes in information, and the transformation
from the underlying scale on which information dissipates to alternative scales
on which probability pattern may be expressed. Information invariances set the
commonly observed measurement scales and the relations between them. In
particular, a measurement scale for information is defined by its invariance to
specific transformations of underlying values into measurable outputs.
Essentially all common distributions can be understood within this simple
framework of information invariance and measurement scale.
