Representations of data and belief

This thesis deals with three types of representational elements: discrete values, continuous values, and relations. This applies to both internal beliefs of the system and observations (or data). Other types of data should be possible to convert to them in a more or less sensible manner. Anderberg (1973) discusses the representations in detail, excluding relations.

For categorical variables, only a finite number of values is possible. For instance, the blood type of a person is one of four possibilities. A coin toss has two possibilities. The alphabet has 26 letters. Text is often processed by giving a discrete label to each known word.

The measured sound pressure in a room is an example of a continuous value or a real number. Most physical measurements come as continuous values, such as measuring the time, weight, length, or temperature. Also the sensory systems in living organisms and robots produce continuous values. Digital cameras and scanners convert images into data where the image is divided into small square pixels that have a constant colour. The colours are described by three numbers, the red, green, and blue intensities. Sound waves can be represented as a sequence of air pressure values, like in CDs. Note that even though values are always represented with limited accuracy in computers, in theory they are handled as real numbers.

Discrete numerical variables, such as the number of children, can be processed as categorical data by making a finite number of categories such as 0,1,2,3,4,5+. The other option is to reinterpret the ordinal value as a continuous value. This is often done by people, too. We have no difficulties in understanding a statement such as ``Finnish women give birth to 1.7 children on average''. Section IV B of Publication III studies and solves a problem originating from a conversion of discrete to continuous values.

The third type of representation, relations, is rather different from the other two. Relations are used to relate objects to one another. Codd (1970) wrote a significant paper about general relational databases. The basic idea is that access to the data is unaffected by the internal representation. This becomes important when more and more different types of data are integrated together into a common databank. Relational databases have become a standard. The universal model for data is basically a set of tables where different columns are different attributes that can be of varying type, and rows are objects. Values in the table may include identifiers that point to other rows of the same or another table. For example, a molecule can be represented as two tables, where the first one lists all atoms with their identifiers and attributes, and the second table lists all bonds, with identifiers of the involved atoms and the attributes of the bond. Similar representation applies to web pages, where instead of bonds, the second table lists links.

As the biological senses never produce identifiers directly, they have to be created by the mind. For example when a predator tries to catch its pray by wearing it down, it is important for the chaser to stick to the same target even if it cannot be recognised from the herd. Also, to know the structure of a molecule, one has to know which atoms are connected with bonds even if the atoms as such are indistinguishable. Both cases can be solved by giving an implicit or explicit identifier for the prey or atom. Pointers are the identifiers used in computer code to refer to different parts of the memory, and thus to different objects.

In Bayesian analysis, the belief or uncertainty about variables is represented with probabilities. The probability of an event $A$ given prior knowledge $B$ is written as $P(A\mid B)$. Similar notation can be used when $A$ is a discrete variable: $P(A\mid B)$ denotes the probability distribution of $A$ given $B$. Continuous probability distribution can be represented with a probability density function (pdf) $p(\cdot)$. The actual probability is an integral over the pdf. It is also called probability mass, using an analogy from physics. For example the probability of an event $A<0$ given $B$ can be computed as $P(A<0\mid B)=\int_{-\infty}^0 p(A\vert B)dA$.

The rest of the chapter is written for continuous values, but rewriting the integrals as sums produces the corresponding formulas for discrete values. The treatment of relations is left to Chapters 5 and 6.