Recursive Bayesian estimation is an approach used in statistics and machine learning (ML) that estimates the current state of a system. The framework is used in robotics and automotive technology, where machines are taught to perform a task that requires estimation.

A self-driving car, for example, can estimate its location using the recursive Bayesian estimation framework. The car obtains its starting position via a Global Positioning System (GPS). Its built-in algorithms then help it estimate its current location after a certain amount of time or distance. The algorithms often use the Bayesian mathematical concept in statistics.

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Understanding the recursive Bayesian estimation framework requires familiarity with two concepts—recursive estimation and Bayesian inference. These concepts are explained in simple terms below.

Recursive Estimation

Estimation is something we do all the time. For example, we can estimate the time it takes to travel from our home to the grocery store based on experience and how we plan to get there. In mathematics and science, researchers refer to this estimated travel time as the “state.”

A state refers to the entities that we care about. It could be travel time, the location of an object, the temperature of a person, or the position of a vehicle, among others.

Recursive estimation refers to the process of estimating the current state based on the previous state. The new state estimation will become the basis for the subsequent state estimation. This method is similar to how we estimate the location of a parcel from Amazon based on the details of previous tracking updates. However, recursive estimation in statistics and ML involves complex mathematical operations. One such operation is Bayesian inference.

Bayesian Inference

Bayesian inference is a statistical method based on Bayes’s theorem, which states that the probability of a state can be based on data and previous beliefs about the state. For example, if we were to predict a basketball player’s probability of making the next shot successfully, we may base our prediction on the player’s distance from the ring. The farther the player is from the ring, the less probable that he can make the shot.

However, Bayesian inference tells us to also take into account previous data. If the player already had six successful three-point shots and four unsuccessful ones, then that translates to:

  • 60% probability of a successful shot 
  • 40% probability that he won’t make the shot

Bayesian inference combines the two methods of making predictions discussed above using a mathematical formula that looks something like this:

P(A|B) = [P(B|A) * P(A)] / P(B)


  • P = the probability of the event occurring
  • A and B are certain events
  • P(A) = probability of event A occurring
  • P(B) = probability of event B occurring

Recursive Bayesian Estimation Example

When used in recursive estimation, Bayesian inference makes it possible to obtain the state of an entity based on two things:

  • The prior state of an entity
  • The likelihood function or prediction

In the basketball example we previously cited, these considerations are:

  • Prior state of an entity: Past attempts of the player.
  • Likelihood function or prediction: Uniform distribution probability dictated by the player’s distance from the ring.

Hence, the prediction may look like this when plotted on a graph:

Recursive Bayesian estimation applications range from the automotive industry to medical science. Self-driving cars use algorithms based on this mathematical approach. The remaining useful life of medical devices can be predicted using recursive Bayesian estimation as well. Industrial robots are designed to estimate their position using this estimation method, too.
Recursive Bayesian estimation is one of the most essential mathematical approaches that have helped advance AI technologies.