Mastering Priors and Assumptions: A Layered Guide for Effective Mathematical Modeling

Mathematical modeling uses mathematical language and methods to describe, analyze, and solve real‑world problems. Building a model requires simplifying and abstracting complex phenomena, which inevitably involves prior knowledge and assumptions.

Good priors and assumptions make models more reasonable, concise, and improve predictive ability and practical value.

This article shares an understanding of the hierarchy of priors and assumptions to help construct and evaluate models.

What Are Priors and Assumptions?

In mathematical modeling, priors refer to knowledge known before modeling, such as scientific laws, domain experience, expert insight, or historical data. Priors are the starting point that helps determine the model’s direction and basic structure.

Assumptions are simplifications or abstractions of reality. Introducing assumptions allows us to ignore unimportant or unobservable factors, making the problem easier to handle. The reasonableness of assumptions directly affects model accuracy and applicability.

Priors and assumptions complement each other: we usually propose a series of assumptions based on prior knowledge, then build the mathematical model on those assumptions.

Levels of Priors and Assumptions

During the modeling process, priors and assumptions can be divided into different levels according to importance, scope, and impact on results.

1. Basic‑Level Assumptions

Definition: Basic‑level assumptions are the most fundamental premises; if they fail, the entire model collapses. They are usually based on scientific laws or widely accepted common sense.

Example: In classical physics, Newton’s second law is a basic‑level assumption for many mechanical models, implying constant mass and a linear relationship between force and acceleration. In modeling, similar basic assumptions might be “population size is a positive integer” or “market is driven by supply and demand.”

2. Modeling‑Level Assumptions

Definition: These are simplifications or approximations introduced to make the problem easier to handle, often derived from priors.

Example: In biology, one might assume a population’s carrying capacity is constant, even though it can change over time. In economics, the “rational agent” assumption simplifies analysis despite real human behavior being imperfectly rational.

3. Data‑Level Assumptions

Definition: Assumptions about the data used in modeling, covering source, quality, distribution, and correlation.

Example: In machine learning and statistics, the IID assumption states that data points are independent and identically distributed. In time‑series analysis, one often assumes noise follows a normal distribution.

4. Result‑Level Assumptions

Definition: After a model is built, additional assumptions may be introduced for analysis or prediction; they do not affect structure but influence outcomes.

Example: A financial risk model may assume future market volatility resembles the past, or an optimization problem may assume the objective function is convex.

Reasonable Use of Priors and Assumptions

Using priors and assumptions wisely is key to building effective models. Recommendations:

Clarify the purpose: Determine whether the model aims to describe phenomena, explain causes, or make predictions, and choose appropriate priors and assumptions.

Leverage prior knowledge: Use domain experts’ insights and existing scientific theory to shape model structure and hypotheses.

Introduce assumptions cautiously: Validate and verify assumptions, especially those with large impact, through data or experiments.

Layered testing and revision: After building the model, test each level of assumptions; revise or reconstruct the model if any assumption fails.

Consider the impact of assumptions: When applying the model, discuss how assumptions affect results and document them in reports.

Example: Priors and Assumptions in the SIR Epidemic Model

The classic SIR model divides a population into Susceptible (S), Infected (I), and Recovered (R) groups and describes their dynamics with differential equations.

1. Basic‑Level Assumptions

The total population size is constant.

The disease spreads only through person‑to‑person contact, with no external inputs.

Violating these assumptions makes the model inapplicable.

2. Modeling‑Level Assumptions

Each individual has an equal probability of contacting any other individual.

Recovered individuals cannot be reinfected.

These simplify the model but may not hold in reality.

3. Data‑Level Assumptions

The data accurately reflect the true number of infections.

The infection and recovery rates are fixed.

In practice, rates often vary over time.

4. Result‑Level Assumptions

Predictions may assume the epidemic will peak and then subside, though real outbreaks can be influenced by many unpredictable factors.

Correctly understanding and applying priors and assumptions at different levels enables the construction of more effective and useful models. Modelers should stay sensitive to assumptions, continuously test and adjust them to ensure accuracy and applicability.

Hope this article helps you better grasp the hierarchy of priors and assumptions in mathematical modeling and apply them flexibly in practice. (Author: Wang Haihua)