A probabilistic model is a mathematical description of an uncertain situation. It must be in accordance with a fundamental framework which has two main ingredients:
Introduction
A probabilistic model is a quantitative description of a situation, a phenomenon, or an experiment whose outcome is uncertain. Putting together such a model involves two key steps.
First, we need to describe the possible outcomes of the experiment. This is done by specifying a so-called sample space Ω.
Second, we specify a probability law, which assigns probabilities to outcomes or to collections of outcomes. The probability law tells us, for example, whether one outcome is much more likely than some other outcome.
Probabilities have to satisfy certain basic properties in order to be meaningful. These are the axioms of probability theory. For example probabilities cannot be negative. Interestingly, there will be very few axioms, but they are powerful, and we will see that they have lots of consequences. We will see that they imply many other properties that were not part of the axioms.
Sample space and Events
Every probabilistic model involves an underlying process, called the experiment, that will produce exactly one out of several possible outcomes. The set of all possible outcomes is called the sample space of the experiment, and is denoted by Ω. A subset of the sample space, that is, a collection of possible outcomes, is called an Event. It is important to note that in our formulation of a probabilistic model, there is only one experiment.
The sample space of an experiment may consist of a finite or an infinite number of possible outcomes. Finite sample spaces are conceptually and mathematically simpler. Still, sample spaces with an infinite number of elements are quite common. As an example, consider throwing a dart on a square target and viewing the point of impact as the outcome.
Regardless of their number, different elements of the sample space should be distinct and mutually exclusive, so that, when the experiment is carried out, there is a unique outcome.
Generally, the sample space chosen for a probabilistic model must be collectively exhaustive, in the sense that no matter what happens in the experiment, we always obtain an outcome that has been included in the sample space. In addition, the sample space should have enough detail to distinguish between all outcomes of interest to the modeler, while avoiding irrelevant details.
To summarize– this set should be such that, at the end of the experiment, you should be always able to point to one, and exactly one, of the possible outcomes and say that this is the outcome that occurred. Physically different outcomes should be distinguished in the sample space and correspond to distinct points. But when we say physically different outcomes, what do we mean? We really mean different in all relevant aspects but perhaps not different in irrelevant aspects.
Probability Laws
Suppose we have settled on the sample space Ω associated with an experiment. To complete the probabilistic model, we must now introduce a Probability Law.
Intuitively, a probability law specifies the “likelihood” of any outcome, or of any set of possible outcomes (an event, as we called it early). More precisely, the probability law assigns to every event A, a number P(A), called the probability of A, satisfying the following axioms:
1. Nonnegativity.
2. Additivity. If A and B are two disjoints events, then the probability of their union satisfies the following:
More generally, if the sample space Ω has an infinite number of elements and A1, A2, A3, A4,… is a sequence of disjoint events, then the probability of their union satisfies:
3. The probability of the entire sample space Ω is equal to 1, that is:
In order to visualize a probability law, consider a unity of mass which is “spread” over the sample space Ω. Then, P(A) is simply the total mass that was assigned collectively to the elements of A. In terms of this analogy, the additivity axiom becomes quite intuitive: the total mass in a sequence of disjoint events is the sum of their individual masses.
There are many natural properties of a probability law which can be derived from them. For example, using the normalization and additivity axioms we may find out the probability of the empty event P(Ø) as following:
This implies that:
Discrete Model - Discrete Probability Law
If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element. In particular, the probability of any event {s1, s2, …., sn} is the sum of the probabilities of its elements:
In the special case where the probability P(s1), P(s2), …, P(sn) are all the same, in view of the normalization axiom we obtain the following law.
Discrete Uniform Probability Law
If the sample space consists of n possible outcomes which are equally likely (i.e., all single-element events have the same probability), the probability of any event A us given by:
Continuous Model
Discrete models are conceptually much easier. Continuous models involve some more sophisticated concepts.
Probabilistic models with continuous sample space differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterize the probability law.
Properties of Probability Laws
Probability laws have a number of properties, which can be deduced from the axioms. Some of them are summarized below:
The role of Probability Theory
Probability theory can be a very useful tool for making predictions and decisions that apply to the real world. Now, whether your predictions and decisions will be any good will depend on whether you have chosen a good model. Have you chosen a model that’s provides a good enough representation of the real world? How do you make sure that this is the case? There’s a whole field, the field of statistics, whose purpose is to complement probability theory by using data to come up with good models. And so we have the following diagram that summarizes the relation between the real world, statistics, and probability. The real world generates data. The field of statistics and inference uses these data to come up with probabilistic models. Once we have a probabilistic model, we use probability theory and the analysis tools that it provides to us. And the results that we get from this analysis lead to predictions and decisions about the real world. Suggested video: Interpretation and uses of Probability
Sources:
- Introduction to probability (bertsekas, 2nd, 2008)
- Probability – The Science of Uncertainty and Data (MITx – 6.431x)
Literature review made by:
Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer
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