# Probability model vs statistical model vs stochastic model

I understand that a statistical model is a model which accounts for the uncertainty in the model. Eg. Demand price equation: $$demand_{i}= a+b price_{i}+ u_{i}$$ where $$u_{i}$$ refers to the residual term.

How to differentiate the other two models from a statistical model?

Many times there is observable phenomena that is random in nature. We call it a Random Process (Random Experiment). The random process has outcomes, and subsets of these outcomes are called events. We map these events to a numeric form using Random Variables.

We study and capture our knowledge about this random process by creating a Stochastic Model. The stochastic model predicts the output of an event by (1) providing different choices (of values of a random variable) AND (2) the probability of those choices. These two elements are summarized as a Probability Distribution.

This distribution has some parameters (like mean, standard deviation) which were inferred from the observable phenomena using Statistical Inference.

Before inference, the distribution had unknown (not inferred yet) parameters. It was, hence, a family of distributions, since each value of the parameter is a different distribution. This family is called a Statistical Model.

Usually a statistical model is guessed (exponential, binomial, normal, uniform, bernoulli, etc) using Exploratory Data Analysis, then its parameters are inferred (estimated) by applying statistical inference (say, algorithms involving loss function minimization) to arrive at a stochastic model (statistical model with known parameters) (a.k.a. Estimator) that captures our knowledge about the random process.

The term 'Probability Model' (probabilistic model) is usually an alias for stochastic model.

References:

1 Using statistical methods to model the fine-tuning of molecular machines and systems Steinar Thorvaldsen, Ola Hossjer

 Statistics (Point Estimation) - Lecture One Charlotte Wickham - Berkeley 