What is the difference between the probability model and the statistical model?

By / November 13, 2021

Statistics uses probability theory to construct the statistical models to deal with data and learn distribution and parameters. However, there are few differences between the statistical model and probability model. As a consequence, we can’t use these two models interchangeably.

Table of Contents hide
1 Probability Model
2 Statistical model
3 Differences between a statistical model and a probability model
4 References
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Probability Model

A probability model is a mathematical representation of a random phenomenon. It consists of the triplet (\Omega, \mathbb{F}, \mathbb{P}) where  \Omega is the sample space, \mathbb{F} is a measurable set or a collection of the events within the sample space, and   \mathbb{P} is a probability measure on \mathbb{F} .

  • The sample space (\Omega for a probability model is the set of all possible outcomes.
  • A measureable set or a collection of events  \mathbb{F} is a subset of the sample space \Omega .
  • \mathbb{P} put a numerical value or probabilities assigned to a given events \mathbb{F}.

A probability measure is a real-valued function that is defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and other more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

  • A real-valued function is a function, whose values are real numbers or in other words, it is a function that assigns a real number to each member of its domain.

For example, a probability model can be interpreted as a known random variable X. Suppose it is considered as a Normally distributed random variable with mean 0 and variance 1. In this case the probability measure \mathbb{P} is associated with the Cumulative Distribution Function (CDF)  \mathbb{F} through

F(x)={\mathbb P}(X\leq x) = {\mathbb P}(\omega\in\Omega:X(\omega)\leq x) =\int_{-\infty}^x \dfrac{1}{\sqrt{2\pi}}\exp\left({-\dfrac{t^2}{2}}\right)dt.

If we repeatedly saw independent and identically distributed for X, we could say that, on average, on the large data limit, it follows a normal distribution. The meaning of X follows a Normal distribution is equivalent to saying that it has a Normal distribution.

In probability theory, when a random variable is considered a discrete random variable, the measure function is called Probability Mass Function (PMF). On the other hand, when it is considered a continuous random variable, it is called Probability Density Function (PDF). These PMF and PDF may follow a certain kind of probability distribution.

  • Probability distribution means there is a probability for every value of a random variable. The values of the random variable, along with the respective probabilities, are called a probability distribution. It describes the dispersion of the values of a random variable. Consequently, the kind of variable determines the type of probability distribution. 
  • A random variable is a mapping function from elementary events to a real number. If a random variable is X then it can be represented as X:\Omega \Longrightarrow R.
  • An event is a specific subset of the sample space to which a probability may be assigned. The probability is assigned via the random variable. It takes an event and produces output as an integer number called a probability of an event.

In summary, a probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities.

Statistical model

A Statistical Model is a set S of probability models, and it is a set of probability measures/distributions on the sample space \Omega. The set of probability distributions is usually selected for modeling a certain phenomenon from which we have data. In this case, the parameters and the distribution that describe a certain phenomenon are both unknown as compared to the probabilistic model, where the parameters and the distribution are known.

A statistical model is a particular class of mathematical models. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values but instead have probability distributions; i.e., some of the variables are stochastic. Statistical models are often used even when the physical process being modeled is deterministic.

For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process).

There are three purposes for a statistical model, according to Konishi & Kitagawa.

  • Predictions
  • Extraction of information
  • Description of stochastic structures

Differences between a statistical model and a probability model

In a probability model, we know exactly the probability measure, i.e., PDF or PMF and CDF of distribution and their parameters. However, in a statistical model, we consider a set of distributions without knowing the parameters. The only input we have is data. From data, we need to find distribution and measure function.
It also answers the question of how probability and statistics are related to each other. To do statistical modeling, we need to be familiar with probability models.

References

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