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Bayesian inference


Bayesian inference models the process of learning. That is, we start with a certain level of belief, however vague, and through the accumulation of experience, our belief becomes more fine-tuned. Some people take a dislike to Bayesian inference because it is overtly subjective and they like to think of statistics as being objective. We don't agree, because any statistical analysis is necessarily subjective as a result of the need to make assumptions, but also because many approximations are accepted without question, even without warning that they have been made. For that reason we appreciate the extra transparency of Bayesian inference, but it also frequently provides answers where classical statistics cannot. Perhaps more importantly for us as risk analysts, Bayesian inference encourages our clients to think about the level of knowledge they have about their problem, and what that means to them.


Bayesian inference is an extremely powerful technique, based on Bayes' Theorem (sometimes called  Bayes' Formula), for using data to improve one's estimate of a parameter. There are essentially three steps involved:


  1. Constructing a confidence distribution of the parameter before analyzing the new data set. This is called the prior distribution;

  2. Find an appropriate likelihood function for the observed data; and

  3. Modify the prior distribution using the likelihood function to get a revised estimate known as the posterior distribution.

Gliffy Diagram
displayNameBayes. Inference
nameBayes. Inference

This section starts with some introductory topics:


Introduction to the concept and some simple examples


How to determine likelihood functions


We then turn to the actual execution of a Bayesian inference for risk analysis models with Crystal Ball by looking at various techniques for arriving at the posterior distribution: 

Construction method

Conjugate prior method


Markov Chain Monte Carlo method 

Most important of all, we offer a number of worked examples:


Examples of Bayesian inference calculations


General estimation problems

Identifying a weighted coin


A simple construction example that shows how we use data that describe being above or below a threshold, instead of exact observations


Taylor series approximation to a Bayesian posterior distribution


Example of a Taylor series expansion


Two common statistical problems


A standard statistics problem with the same outcome as the classical method 

Bayesian estimate of the mean of a Normal distribution with known standard deviation

Another standard statistics problem with the same outcome as the classical method 

Once you have reviewed the material in this section, you might like to test how much you have learned by taking the self-test quiz: 

A quiz on Bayesian inference: 

Take Quiz Button
quizA Quiz on the Bayesian Inference
author-full-nameElnaz Beirami