Bayes Theorem, the Geometry of Changing Beliefs
Image created by Holly Burns

 

Bayesian thinking is successfully applied to professional fields requiring statistics and also to those considered outside of traditional statistics like education, computer programming, software, philosophy, economics, medicine and the practice of law. Bayesian thinking is a mental model which allows us to adapt our thinking reactively to new evidence as it arises and adjust our action to optimize the odds of success as the probabilities evolve in real time.

Inside probability theory, conditional probability is a way to calculate and measure the probability of some event happening if another event has already occurred.

The Bayes’ Theorem is one way of calculating a probability of something occurring when you know probabilities of other things happening.

The Bayes’ theorem is a mathematical formula that explains how to update current probabilities of an event happening based on a theory when given evidence of the potential occurrence. It is calculated from the principles of conditional probability, it can be used as a tool for reasoning what could happen after the changing probabilities of a large range of new circumstances that create belief updates.

How do I change my beliefs on Bayes?

As the first step in your process you start with your hypothesis. You set the odds for your hypothesis being true based on your reasoning and logic for your theory.

The second step is that you observe the evidence to see how your hypothesis holds up when confronted with reality and the data.

As the third step in the process you restrict your views about your hypothesis only to what parts of it are confirmed by the evidence of data. You discard anything that is shown to be untrue inside the framework. You limit your new hypothesis to within the framework that is shown to be true.

When making decisions going forward you are looking to calculate the likelihood of your current hypothesis being true based on the evidence you can observe. This gets into the math of calculating probabilities and this is a very different mental model than what most people do simply being guided by their own emotions when it comes to life, investing, trading, sports betting, sports, or business.

The posterior probability is a type of conditional probability that results from updating the prior odds with information of the new updated likelihood, through the use of Bayes’ theorem. The posterior probability contains everything there is to know about an uncertain proposition like a scientific hypothesis, or parameter values, given prior knowledge and a mathematical model describing the observations available at a set time. After new information is received, the current posterior probability may serve as the prior input in a new round of Bayesian updating. [1]

We can think of Bayes rule in terms of updating our belief about a hypothesis (A) when there is new evidence (B). Our previous held belief (posterior) P(A|B) is calculated by multiplying our prior belief P(A) by the likelihood P(B|A) that B will occur if A is true. [2]

What are benefits of Bayes Theorem?

Bayesian thinking allows you to both quantify and systematize the ability to change beliefs. It’s like running your mind with new software. It’s a very powerful mental model to use in any field that requires an edge in statistics or flexible action.

The advantage of using Bayesian analysis includes the following:

  • It allows for reactivity to a situation in real time versus stubbornness.
  • You can follow trends as they evolve.
  • You can identify where you’re wrong and change course.
  • It creates open-mindedness to new feedback.
  • Bayesian thinking creates flexibility in action.
  • It provides a principled strategy of combining previous information with new data, within a framework.
  • You can combine past information about a specific parameter and create a new analysis.
  • It allows for input of new data into existing formulas.

Bayes theorem is a framework for decision making based not on predictions, opinions, or ego but the evolving statistical analysis of incoming data filtered through an existing model.

A Bayesian conclusion uses an estimate of the odds to come up with the degree of belief in a hypothesis or theory before new evidence has been analyzed and then also calculates the estimated odds or the degree of belief in the hypothesis after each new evidence has been observed. This process of observing, thinking, and resetting the probabilities is repeated each time additional new evidence is obtained. [3]

Bayesian thinking with decision making involves basing decisions on the probability of a successful outcome, where this probability is set by both prior information and new evidence the decision maker receives. The statistical analysis that underlies the calculation of these probabilities is Bayesian analysis. [4]

Bayesian psychology is strong in its original framework of theory but flexible in its adjustments of beliefs of outcomes based on new and evolving inputs. Bayesian thinkers see the world as a flow chart of odds and probabilities being constantly updated with new data that sets and resets trajectories.

A Bayesian thinker looks at the math and is constantly recalculating the odds and doesn’t try to project out too far in the future as they know all the complexity and moving parts involved in events in real time until they reach their final outcome.

The language of a Bayesian is math, probabilities, and open-mindedness. They see the world in statistics not set outcomes. Bayesian thinking is statistical reasoning that is in flux and always updating.

Bayes Theorem Formula

Below is Bayes’ theorem formula expressed as a mathematical equation:

where  and  are events and .

  •  is a conditional probability: the likelihood of event  occurring given that  is true.
  •  is also a conditional probability: the likelihood of event occurring given that  is true.
  •  and  are the probabilities of observing  and  independently of each other; this is known as the marginal probability.

The Bayes Theorem can be used to calculate the probabilities of something happening in conditions where multiple other things are occurring. It’s one mathematical model that can be used for managing a current trade as it plays out in real time, the thinking behind using entry signals, stop losses, trailing stops, and profit targets is Bayesian thinking.