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[Legion]

Being neither mathematician or scientist, I must confess that I am a layman. However, what follows herein is hopefully known as a linguistic exploration and artifice . It is hoped to be moving towards a system of implication which is held in the imagination and communicated with others through the use of symbols. In striving to be a formal language construct its uses may include, but need not include, the potential to mirror causal relations. If it has any utility, then some of it may reside in the potential to allow phenomenal representations and inferences within it and thereby allow for the fabrication of hypotheses. In order to suggest that many bridges may exist between the languages of statistics and categories, I hope to provide a rough outline here of one such bridge.
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Suppose we have before us some handful of events which we are able to represent as a set, and we will name this set, E. Further suppose that there are two events in this set named a and i, E={a, i}. We will also imagine that circumstances conspire to allow transformative paths to exist between these events such that each event may be transformed into itself and into the other event. And the set of these transformations will be denoted as T. We will also entertain that each transformative path has a number on the unit interval, U = (0, 1), assigned to it which indicates the probablity of this transformation. We are thus supposing a graph such as something shown in figure 1.

 statistical category.jpg   10.56K   2 downloads

Thus, so far, we may express this graph in categorical terms as three sets and three maps. Our sets are: the set of events E={a, i}, our set of transformations T={w, x, y, z}, and the unit interval U=(0,1). And our maps are: a map s:T-->E which gives the souce event of each transformation, a map b:T-->E which gives the target event of each transformation, and a map p:T-->U which gives the probability of each transformation.

Some of the thoughts which occur to me under this view are...

If the probability of all transformations sharing the same source must have a sum of 1, then how is this expressed categorically?

If when given any event as a source, we produce the event which is arrived at by following only the most probable transformation at each source event, then doesn't this produce a dynamic system, E-->E?

There are no probabilities assigned directly to each event, but there seem to be implied probabilities associated with them by virtue of the fact that transformations to them are statistically favorable.

This completes a first attempt to examine statistics in the language of category theory.

Edited by Legion, 23 June 2012 - 01:22 PM.