Model-based inference is a statistical approach that involves the use of mathematical models to make predictions or inferences about a given set of data. This method is often used in fields such as economics, engineering, and biology to analyze complex systems and make predictions based on the data.

One example of model-based inference is the use of linear regression to predict the value of a dependent variable based on the value of one or more independent variables. In this case, the mathematical model is a linear equation that describes the relationship between the dependent and independent variables. By fitting the data to this model, we can make predictions about the value of the dependent variable based on the values of the independent variables.

Another example of model-based inference is the use of Markov chain models to predict the probability of certain events occurring. In this case, the mathematical model is a set of equations that describe the likelihood of transitions between different states in a system. By fitting the data to this model, we can make predictions about the probability of certain events occurring, such as the likelihood of a customer purchasing a product based on their previous purchases.

Overall, model-based inference is a powerful tool for making predictions and inferences about complex systems. It allows us to take into account multiple variables and the relationships between them, and make more accurate predictions than we could with other statistical methods.