eikosograms - The Picture of Probability
An eikosogram (ancient Greek for probability picture)
divides the unit square into rectangular regions whose areas,
sides, and widths, represent various probabilities associated
with the values of one or more categorical variates. Rectangle
areas are joint probabilities, widths are always marginal
(though possibly joint margins, i.e. marginal joint
distributions of two or more variates), and heights of
rectangles are always conditional probabilities. Eikosograms
embed the rules of probability and are useful for introducing
elementary probability theory, including axioms, marginal,
conditional, and joint probabilities, and their relationships
(including Bayes theorem as a completely trivial consequence).
They are markedly superior to Venn diagrams for this purpose,
especially in distinguishing probabilistic independence,
mutually exclusive events, coincident events, and associations.
They also are useful for identifying and understanding
conditional independence structure. As data analysis tools,
eikosograms display categorical data in a manner similar to
Mosaic plots, especially when only two variates are involved
(the only case in which they are essentially identical, though
eikosograms purposely disallow spacing between rectangles).
Unlike Mosaic plots, eikosograms do not alternate axes as each
new categorical variate (beyond two) is introduced. Instead,
only one categorical variate, designated the "response",
presents on the vertical axis and all others, designated the
"conditioning" variates, appear on the horizontal. In this way,
conditional probability appears only as height and marginal
probabilities as widths. The eikosogram is therefore much
better suited to a response model analysis (e.g. logistic
model) than is a Mosaic plot. Mosaic plots are better suited to
log-linear style modelling as in discrete multivariate
analysis. Of course, eikosograms are also suited to discrete
multivariate analysis with each variate in turn appearing as
the response. This makes it better suited than Mosaic plots to
discrete graphical models based on conditional independence
graphs (i.e. "Bayesian Networks" or "BayesNets"). The
eikosogram and its superiority to Venn diagrams in teaching
probability is described in W.H. Cherry and R.W. Oldford (2003)
<https://math.uwaterloo.ca/~rwoldfor/papers/eikosograms/paper.pdf>,
its value in exploring conditional independence structure and
relation to graphical and log-linear models is described in
R.W. Oldford (2003)
<https://math.uwaterloo.ca/~rwoldfor/papers/eikosograms/independence/paper.pdf>,
and a number of problems, puzzles, and paradoxes that are
easily explained with eikosograms are given in R.W. Oldford
(2003)
<https://math.uwaterloo.ca/~rwoldfor/papers/eikosograms/examples/paper.pdf>.
