# Plausibility, Possibility, and Probability

When we talk about probability, are we referring to the Plausibility in term of numbers? Or are we referring to the possibility in term of numbers? In statistics (or Mathematics), is there a difference between these two (plausibility &possibility)?

### Possibility and probability can be examined together

This answer is based on material in the paper O'Neill (2014) which gives a detailed exposition of the relationship between probability and possibility for the assistance of statistics students and practitioners. Some of the material in this answer is copied from that source without further citation/attribution. I recommend you have a look at that paper to get a more detailed exposition of the material I discuss in this answer. You can also find a related answer here.

I am not aware of any formal definition of "plausibility" beyond observing that it is sometimes used as a quantification of probability. However, I can tell you the relationship between the other two concepts. Possibility is a slightly different concept than probability, which demarcates all the outcomes that can occur within a wider space. There is an alternative algebra for "possibility measures", with a different kind of additivity property that reflects the fact that the possibility measure is "compositional". A possibility measure can be applied over the set of all events on an arbitrary set of outcomes, but if the class of events has the required structure for a probability measure, then we can also have a probability measure over the same class of events, which allows us to examine the interaction of the two measures. The two measures interact on the basis of a simple and intuitive rule: every impossible event has zero probability. (Note that the converse is not always true.)

There is a large literature looking at mathematical representations of possibility (known as "possibility theory"), which is closely related to fuzzy set theory. Overviews of this literature can be found in Yager (ed) (1982), Kacprzyk and Orlovski (eds) (1987), Dubois and Prade (1988), Terano, Asai and Sugeno (1992) and Zadeh and Kacprzyk (1992). A useful analysis of the relationship between possibility and probability representations can be found in Dubois and Prade (1993).

Formal presentation: Suppose we let $$\Omega_*$$ denote some set of outcomes and let $$\mathscr{G}_*$$ be a class of events with sufficient structure to allow a probability measure (i.e., it is a sigma-field). Suppose we have a possibility measure $$\mathbb{pos}$$ and a probability measure $$\mathbb{P}$$ on this class of events, giving rise to a possibility/probability space $$(\Omega_*, \mathscr{G}_*, \mathbb{pos}, \mathbb{P})$$. This space allows us to examine the interaction between possibility and probability. The starting point is our basic axiom (that I will call the "axiom of correspondence"), which says that every impossible event must have zero probability:

Axiom of correspondence: For all events $$\mathcal{E} \in \mathscr{G}_*$$ we have:$$\mathbb{pos}(\mathcal{E}) = 0 \quad \quad \implies \quad \quad \mathbb{P}(\mathcal{E}) = 0.$$

It is useful to narrow this down to look at only the outcomes that are possible. The set of possible outcomes (which I will call the possibility space) is given by $$\Omega \equiv \{ \omega \in \Omega_* | \mathbb{pos}( \{ \omega \} ) > 0 \}$$, and this gives rise to the subclass of events $$\mathscr{G}$$ on $$\Omega$$. All outcomes outside the possiblity space (i.e., in $$\Omega_* - \Omega$$) are impossible. It is easy to use the properties of the possibility measure, and the above axiom, to show the following:

Events outside the possibility space: For all events $$\mathcal{E} \subseteq \Omega_* - \Omega$$ we have $$\mathbb{pos}(\mathcal{E}) = \mathbb{P}(\mathcal{E}) = 0$$; all these events are impossible and have probability zero.

Once the possibility space is defined, it is possible to rewrite the axiom of correspondence in the more compact form show below. This version states that the possibility space is an almost sure event.

Axiom of correspondence (alternative form): We have $$\mathbb{P}(\Omega) = 1$$.

If you have done a foundational course in probability theory, you will recognise this as the norming axiom of probability, which applies to the "sample space". Thus, this latter version of the axiom of correspondence is the justification for why we would ordinarily just start on the space $$\Omega$$ in applications of probability theory. Ordinarily, we would take this set of possible outcomes to be our "sample space" and we would form the probability space $$(\Omega, \mathscr{G}, \mathbb{P})$$ using that space. In this formulation, every outcome in the sample space is considered to be a possible outcome (since that space is just the possibility space). Any outcomes outside this are impossible, and so they have zero probability, and can safely be ignored.

Since probability theory forms the basis for statistical analysis, this essentially means that we can ignore a set of outcomes with probability zero, and concentrate attention on some sample space that occurs almost surely. Removal of impossible events is a natural part of this reduction.

As you can see, there is a clear formal interaction between the concepts of "possibility" and "probability". This addresses the common confusion that students have when they deal with continuous random variables, where we have outcomes that have zero probability density (and therefore zero probability), but which are nonetheless possible outcomes. If you read some of the literature on possibility theory you will see that it allows the possibility measure to be "fuzzy", in the sense that it can take on values other than zero and one. However, it is instructive to consider the case where this measure is restricted to values of zero or one, to get a basic intuition for the interaction of possibility and probability.