Part of the The Bayesian Investor — A Primer for the CIO series.

Phase 1 — Foundations: Probability, Updating, and the Tools of the Trade

What probability really is

We use the word "probability" constantly, but let's pin it down. A probability is simply a number between zero and one that says how likely something is. Zero means "it won't happen," one means "it definitely will," and everything in between is a shade of maybe. A probability of 0.7 means "about seven times out of ten." That's all.

But there is an important fork in the road that determines which kind of probability you're using, and it matters enormously in investing.

The first kind is frequency probability. It answers questions like "If I flip this coin a million times, how often does it land heads?" It is the probability of repeatable events — the world of coin flips, dice, and long historical records. An actuary counting past insurance claims lives here.

The second kind is belief probability — a measure of confidence about a single, one-off event. "How likely is it that this specific acquisition outperforms its underwriting?" That event happens once. It has no long-run frequency. There is no million-time history of this exact deal to count. And yet you make decisions on it every day. Belief probability is the honest way to put a number on that: it is your degree of confidence, expressed on the zero-to-one scale, informed by whatever you know.

For a CIO, the punchline is immediate: frequency probability is useful, but belief probability is the one your job actually runs on. Every deal, every market call, every regime call is a one-off event. This roadmap is about getting really good at the second kind — while remaining honest that it is a belief, not a fact.

The core move: updating a belief

Now the bookmaker's move, formalized gently.

You begin with a prior — your belief before seeing new evidence. Then evidence arrives. You update. What you're left with is your posterior — your belief after the evidence. And here's the elegant part: today's posterior becomes tomorrow's prior. The loop never ends.

Prior = your starting belief (e.g., "I think this property's cap rate compression has a 40% chance of continuing").
Evidence = what you observe (e.g., the Fed cuts, transaction volume picks up).
Posterior = your updated belief (e.g., "now I think it's a 55% chance").

Three rules of thumb make the whole machinery intuitive, and they're worth internalizing because everything later is a fancier version of them:

Thin evidence, prior rules. When you have almost no data, your starting belief dominates. This is not a bug — it's the honest recognition that with little to go on, your informed judgment is the best estimate you have.
Strong evidence, data rules. As evidence piles up, the data increasingly dominates and your starting belief fades. The bookmaker's early nervousness gives way to a settled price.
Confidence grows with evidence. A belief updated by a hundred data points is firmer than the same belief updated by ten. The width of your uncertainty narrows — you don't just get a better guess, you get a guess you trust more.

That last point is the deepest gift of this way of thinking, and it's the thing most traditional finance tools quietly omit: an honest, ever-present measure of how sure you are. A number without its confidence is a rumor; a number with its confidence is a decision input.

The coin-flip mental model

Your team's first exercise is trivial on its face: simulate coin flips. But it's actually the perfect laboratory for the three rules above.

Suppose you believe a coin is fair — a 50/50 starting belief, a prior. You flip it ten times and get seven heads. Should you abandon your prior? No — ten flips is thin evidence, so your belief moves only a little. Now flip it a thousand times and get seven hundred heads. Now the evidence is overwhelming; your belief should move decisively toward "this coin is biased." And notice the confidence: after ten flips you'd be unsure; after a thousand you'd be certain.

Doing this in simulation — having the computer run tens of thousands of flips at a keystroke — lets your team feel the rules rather than memorize them. It's the difference between reading about how a river moves and standing in it.

Python, in words

Somewhere in this phase your team will start using Python. Here's the only thing you need to understand about it as a decision-maker: Python is a general-purpose calculator that can run millions of imaginary what-ifs in seconds. It lets your analysts describe a belief and a piece of evidence in ordinary language-shaped code, and then have the computer do the heavy arithmetic — arithmetic that would take a human a lifetime, by hand, and is prone to human error besides.

Why the Python ecosystem specifically? Because the world's best Bayesian tooling happens to live in it — free, battle-tested, and shared openly by a huge community of practitioners. Your team isn't choosing a programming language for the sake of programming; they're choosing the workshop where the modern tools of this discipline are kept. Phase 1 gets them comfortable walking around that workshop; the later phases put the tools in their hands.