Update belief by Bayesian Inference

Bayesian Inference is a mathematical method of updating beliefs (or probability) of any uncertainty based on evidence or observation from experimentation, using Baye's theorem. In Decision Tree Software, you can Update your belief/probability of Chance node events using Bayesian Inference. A tool for Bayesian Inference is built in for you.

Let's show a worked-out example. Say, you received a job offer and you are in a dilemma if you should take the job or wait for a better offer. The current job offer will pay you 80,000$/year but you think there is a 70% probability that if you wait, you may get a better offer. Should you wait or take the current offer at hand. The dilemma is shown in the following decision tree:

Starting decision tree used for the Bayesian inference example, modelling the choice between accepting a job offer and waiting for a better one.

From the Sensitivity Analysis, you understood that the most sensitive variable in this decision tree is the probability of getting a better offer.

So, now you want to do some experiments and update the belief. You can upload your resume to an online job-finding website (like Monster.com, jobs.com, etc) and apply to various jobs and observe the rate of responses. If you get a good response, you can update your belief of the probability that you will get a better offer would be higher. But if you get very few responses, then your confidence in getting a better offer will be lower.

Let's start by selecting the Chance node and clicking the "Experiment" button from the flyover menu as shown below.

Experiment toolbar button on a chance node, used to open the Bayesian inference tool and update the probability of its uncertain events from evidence.

Once you click the experiment button, you will see the Bayesian inference tool like this:

Bayesian inference tool window opened for a chance node, showing initial beliefs, the lock button, and the controls for adding experiments and observations.

Once you click the Lock button to lock your initial beliefs, start experimentation by clicking the button "Add a new experiment".

Editing the label of a new experiment in the Bayesian inference tool to describe how evidence will be gathered (for example, Check newspaper for job postings).

Then edit the experiment as "Check newspaper for job postings".

Toolbar buttons in the Bayesian inference tool for adding a new experiment and adding an observation under the selected experiment.

Add another experiment as "Apply to some jobs on a job website".

Bayesian inference tool with two experiments added (Check newspaper for job postings and Apply to some jobs on a job website) ready for observations.

Now, check the newspaper and count the number of job postings in your field of interest. Once you get that information, add that information as an observation under the experiment "Check newspaper for job postings"

Add Observation button used to record real-world evidence (such as the count of job postings found) under a Bayesian inference experiment.

Update the observations for the experiments once you perform the experiments.

Bayesian inference tool with observations recorded under each experiment, ready for the causal discovery / belief update step.

Now, it is time to perform a Causal discovery. Causal discovery is a methodology to find a cause based on observation. And here, the software uses Baye's theorem to update the beliefs.

Causal discovery prompt where the user supplies the likelihood of each observation given the hypothesis, feeding Bayes' theorem during belief update.

Now is the time to apply your reasoning skill. What do you think that you will stay jobless when you see 2 job postings? Did you expect more job postings? Then lower the likelihood number for observation "given 'Stay jobless for a long time is true. If you expected fewer job postings, then increase this number. Say, for example, you expected more job postings in the newspaper than 2, so, let's set the likelihood to 80%. Same way, set the likelihood of the observation "given 'Get a better offer' is True = 30%.

Likelihood number input in the Bayesian inference tool used to express how many job postings the user expected versus the count actually observed.

Once you click the Update Belief button, you will be asked a similar question for other experiments. Finally, you will see that your beliefs are updated as shown below.

Updated-beliefs result panel showing the posterior probability of getting a better job offer, dropped from 70 percent down to roughly 11 percent after evidence.

So, you see that your belief of "getting a better offer" changed from 70% to merely 11.11%. Click the Complete button in the Ribbon now.

Complete button in the ribbon used to dismiss the Bayesian inference tool and push the updated posterior probabilities back into the decision tree.

Once you click the complete button, you will see the Bayesian Inference tool got dismissed and the Decision tree Chance node is updated with the new probabilities. And then, you will see that the decision tree Expected values are updated based on the new probabilities and the policy is now changed from "Wait for a better offer" to the action "Take the job offer at hand".

Decision tree after Bayesian inference: the chance node probabilities and expected values are updated, switching the optimal policy to Take the current offer.

So, we see that Bayesian Inference based experimentation can significantly improve your decision quality. It is always recommended to perform sensitivity analysis and use the Bayesian Inference tool to update beliefs about any uncertainty especially for those events that have probability with high sensitivity.