Bayes' Theorem

Bayes' Theorem (also known as Bayes' Rule) is a mathematical formula from probability theory that describes how to rationally update a belief when you are presented with new evidence. For a value investor, this isn't about plugging numbers into a complex equation on a daily basis. Instead, it’s a powerful mental model for disciplined thinking in a world of uncertainty. It provides a formal framework for combining your initial assessment of an investment (your 'prior' belief) with new information as it arrives—such as an earnings report, a new competitor, or a change in management—to form a more refined and logical conclusion (a 'posterior' belief). At its heart, Bayesian thinking is the antidote to stubbornness and emotional reactions. It encourages investors to continuously learn and adjust their views, ensuring that their investment thesis evolves with the facts rather than remaining stuck in the past.

Imagine you're a doctor. A patient comes in with a cough. Your initial thought (your prior probability) might be that there's a 10% chance it's a serious illness and a 90% chance it's just a common cold. This is your starting point. Now, you get new evidence: a lab test comes back positive for a specific marker associated with the serious illness. Bayes' Theorem provides the logical structure to update your initial 10% belief. You must consider:

• How likely was the positive test if the patient really has the serious illness? (The test's accuracy).
• How likely was the positive test if the patient just has a cold? (The test's false-positive rate).

By weighing this new evidence, you can calculate a new, updated probability (the posterior probability) that the patient has the serious illness. It will almost certainly be higher than 10%, but the exact amount depends on the quality of the evidence. This is Bayesian updating in a nutshell: Initial Belief + New Evidence = Updated Belief.

For investors, a stock thesis is a hypothesis, and every news story, financial report, and industry trend is a piece of evidence. Bayesian thinking provides the discipline to process this evidence logically.

Every investment starts with a 'prior' belief. You might think, “Company ABC looks cheap based on its Price-to-Earnings Ratio (P/E Ratio).” This is your starting hypothesis. A Bayesian investor doesn't stop there; they become an evidence-gatherer. Let's walk through an example:

1. Prior Belief: Based on its stable industry and strong balance sheet, you believe there's a 70% chance that Company XYZ is a great long-term investment. This is your prior probability.
2. New Evidence: The company announces it's entering a risky new market. Simultaneously, its closest competitor goes bankrupt.
3. Updating Your Belief: This is mixed evidence. The risky expansion might decrease your confidence, but the competitor's failure might increase it. Instead of guessing, you evaluate each piece of evidence. The competitor's demise is strong evidence supporting XYZ's competitive advantage. The new venture is a risk but could have a huge payoff. After weighing the facts, you might adjust your confidence up to 75%. This new figure is your posterior probability. You haven't blindly held your opinion; you've refined it.

A Bayesian mindset is one of the best defenses against the cognitive biases that sink so many investors.

• Confirmation Bias: This is the tendency to only seek out information that supports our existing beliefs. Bayes' Theorem forces us to ask the opposite question: “How likely is this piece of evidence if my thesis is wrong?” By considering alternative outcomes, you can assess the true power of your evidence and avoid falling in love with your own story.
• Base Rate Neglect: We often get mesmerized by a specific, compelling story and forget to ask, “How often does something like this happen in general?” This 'general frequency' is the base rate. For example, before investing in a speculative biotech stock with a “miracle drug,” a Bayesian thinker would first ask: “Historically, what percentage of biotech firms at this stage actually succeed?” This grounds your expectations in reality, preventing you from over-weighting a single, exciting data point.

While the mental model is what counts, seeing the formula can help solidify the concept. P(A|B) = [P(B|A) x P(A)] / P(B) Let's break this down in investment terms. Imagine our thesis (A) is “This company is undervalued” and the new evidence (B) is “It just released a surprisingly strong earnings report.”

• P(A|B) (Posterior Probability): This is what we want to know. The probability that the company is undervalued (A), given that it released a great report (B). This is our new, updated conviction.
• P(B|A) (Likelihood): If the company is truly undervalued and strong (A), what was the probability that it would produce such a good report (B)? We would expect this to be high.
• P(A) (Prior Probability): What was our initial belief that the company was undervalued (A) before we saw the report? This was our starting thesis.
• P(B) (Marginal Likelihood): What is the probability of any company in this industry posting such a strong report? This helps us contextualize the evidence. If everyone is doing well, the report is less meaningful. If only our company did well, the evidence is much stronger.

You will likely never need to calculate this formula with a pen and paper. Its true value lies in the structured thinking it promotes. It's a formal process for being open-minded yet disciplined. Legendary investors like Warren Buffett and Charlie Munger are masters of Bayesian thinking, even if they don't use the term. They constantly absorb new information, test it against their core principles, and are not afraid to update their beliefs—or even admit they were wrong. By embracing the spirit of Bayes' Theorem, you can move from reactive, emotional decision-making to a more rational, evidence-based process. It teaches you that conviction should not be static; it should be earned and re-earned as the facts evolve.