Bayes' Theorem and the Changing of Your Beliefs
A model for better rationality, avoiding the Bayesian thinking trap, and a multi-billion-dollar engagement machine
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Every decision you make is a gamble—but it's the blind spots, biases and poor information that end up costing you more than just your money.
Your brain is great at protection, with fact-finding often reduced to seeking information that confirms beliefs you already have about the world around you.
People tend to avoid questioning their decisions, especially poor ones. But’ poor ones. But what if there were a framework you could use that would do more than swing the odds back in your favour—it could give you the confidence to start making those tough decisions you've been putting off out of fear you'd get them wrong? It turns out the answer is rooted in one of my least favourite high school subjects—maths—and is called Bayesian thinking.
What is Bayesian Thinking?
When you say you believe something to be true, you are engaging in epistemic rationality. But how often do your beliefs match that of reality, and how often do your beliefs lead to you making better decisions in service of your goals? It's all well and good to believe something to be true, but it should be more important to believe in something that is true.
Typically, we state things. Binary—yes or no, true or false—but much in life swings like a pendulum with a lot of stuff in between the edges. What Bayesian thinking does is account for those swings by treating them as probabilities. Sometimes something looks to be one thing, but when new evidence becomes available, it can be another. The problem is that as people, once we've made our minds up, we're pretty terrible at wanting to change it. This was one of Charlie Munger's famed human tendencies—the inconsistency-avoidance tendency.
To change your mind about something is seen as inconsistent and somehow means you lacked knowledge about your claim in the first place. But Bayesian thinking, or updating as it can be called, is just that—updating. This simple word switch, from changing to updating, should be used to combat cognitive biases. You update your phone, your computer, your house, car and life, so why wouldn't you update your thinking?
So, how does it work?
Bayesian Thinking as a Decision-Making Tool
Bayesian thinking offers a systematic framework for updating "degrees of belief" that combines prior knowledge with new evidence when it becomes available to attach a new likelihood (posterior) of that thing happening. It could look like this:
Prior—What we initially believe. "I think there's a 30% chance of delays due to technical issues based on past projects."
Evidence (Likelihood)—The new data or observation. "Given a delay, there's an 80% chance this specific issue shows up."
Posterior—The updated belief after combining prior and evidence. "Now the chance of delay given the issue is 63%."
At the heart of the concept lies Bayes' Theorem:
P(Hypothesis|Evidence) = P(Evidence|Hypothesis) × P(Hypothesis) / P(Evidence)
By assigning a prior belief as a percentage, instead of "I think this" or "I believe that," you can update your thinking and results over time, enabling you to become more efficient at making decisions.
How to Apply Bayesian Thinking Practically
The Five-Step Bayesian Decision Process
First, define your prior beliefs by establishing initial probability estimates based on your previous knowledge, available data, or expert opinion. Second, gather new relevant evidence through data collection, observations, or new information that is brought to your attention. Third, apply Bayes' Theorem to update the probability of your prior now that new evidence is available. Fourth, update your beliefs by replacing your initial priors with these new posterior probabilities. Fifth, make decisions based on your updated understanding, which becomes the foundation for the next cycle.
From the information above, a practical example in words might look like this:
Prior: "30% chance of delay on past projects."
Evidence: "If delayed, issue shows up 80% of the time. If not delayed, it still shows up 20%."
Posterior: "Given the issue, the chance of delay is now ~63%."
Or written in maths:
Prior: P(Delay) = 0.3
P(No Delay) = 0.7
Likelihood: P(Issue | Delay) = 0.8
False positive rate: P(Issue | No Delay) = 0.2
Step 1: Evidence (denominator)
P(Issue) = (0.8 × 0.3) + (0.2 × 0.7) P(Issue) = 0.24 + 0.14 = 0.38
Step 2: Posterior
P(Delay | Issue) = (0.8 × 0.3) / 0.38 = 0.24 / 0.38 = 0.63 or 63% chance of delay given that the specific issue has shown up.
By treating uncertainty as "degrees of belief" that can be quantified and mathematically updated, Bayesian thinking provides a framework that turns guesswork into calculated decision-making.
But there is a trap.
The rest of this post is available to paid members and explains the biggest trap with this framework, tips for mitigating against it, and provides a case study example of a multi-billion dollar company everybody is familiar with using the framework to keep users engaged.