If you want to understand how I forecast the SpaceX IPO price — and how rational people should update their beliefs in the face of new evidence — you don’t need a doctorate in statistics. You just need Bayes’ theorem.
Let’s start with the simplest possible version, and then I’ll drag you deeper.
Part I: The Intuition — Is She Cheating?
You come home. In your wife’s dresser drawer, you find a pair of lingerie you didn’t buy, doesn’t match anything she owns, and in a size that raises questions.
Your brain immediately goes: Is she cheating on me?
Bayes’ theorem answers that question properly — not with a gut reaction, but with arithmetic.
The formula in its cleanest form:
P(A|B) = P(B|A) × P(A) / P(B)
In English: The probability of A given B equals the probability of B given A, times the prior probability of A, divided by the total probability of B.
Naturally nobody thinks in that syntax. Let’s translate it into the lingerie problem.
- A = “My wife is cheating on me”
- B = “I find mysterious lingerie in her drawer”
The question is: P(A|B) — what’s the probability she’s cheating, given the lingerie?
Here’s where most people get it wrong. They jump straight to P(B|A) — “If she were cheating, how likely is it that mysterious lingerie would appear?” That’s obviously high. Let’s say 80%.
But Bayes forces you to confront two other numbers:
P(A) — The prior. Before finding the lingerie, what was your belief she was cheating? If things have been good, maybe 5%. This is the number your gut ignores — your brain leaps from 5% to 90% without doing the math.
P(B) — The total probability of finding lingerie. This isn’t just cheating. She could have bought it for you as a surprise (10%), a friend stayed over (5%), it’s from a photoshoot (2%), or a dozen other explanations. Add those up and P(B) ≈ 35%.
Now do the arithmetic:
P(A|B) = 0.80 × 0.05 / 0.35 = 0.114, or about 11%
So the mysterious lingerie bumps your suspicion from 5% to 11%. That’s a doubling — but it’s still less likely she’s not cheating. Your gut may have screamed “betrayal!” but the math says: breathe and ask her calmly.
This is the entire point of Bayes: new evidence shifts your belief proportionally, not catastrophically. The prior matters enormously.
Part II: Sequential Updating — What If She Also Has New Perfume?
Now complicate it. Two days later, you notice she’s wearing a perfume she didn’t own before.
A is still “she’s cheating.”
B’ = “She has new perfume.”
Your prior is now 11% — not the original 5%. You update from where you left off.
- P(B’|A) = If she were cheating, how likely is new perfume? Say 60%.
- P(B’) = Total probability of new perfume with or without cheating: she might have just bought herself a new scent (40%) or other reasons. Let’s say P(B’) ≈ 45%.
P(A|B’) = 0.60 × 0.11 / 0.45 = 0.146, or about 15%
Another bump, but not the nuclear revelation your emotions promised. Each piece of evidence updates you — gradually, proportionally. This is Bayesian inference in motion: a running estimate, not a verdict.
Part III: The Mathematical Machinery
Now let’s go proper.
Bayes’ theorem is a direct consequence of the definition of conditional probability. For any two events A and B with P(B) > 0:
P(A|B) = P(A ∩ B) / P(B)
Similarly, P(B|A) = P(A ∩ B) / P(A), so P(A ∩ B) = P(B|A) × P(A).
Substituting:
P(A|B) = P(B|A) × P(A) / P(B)
That’s the discrete form. It’s cute, but useless for continuous problems — and most forecasting, including stock price prediction, is continuous.
Part IV: Continuous Variables and the Likelihood Function
When A is a continuous parameter θ (say, the SpaceX IPO price), we replace probabilities with probability density functions. The theorem becomes:
f(θ|D) = f(D|θ) × f(θ) / f(D)
Where:
- f(θ) is the prior density — your belief about θ before seeing data
- f(D|θ) is the likelihood function — the probability of observing data D if the true parameter is θ
- f(θ|D) is the posterior density — your updated belief
- f(D) is the evidence or marginal likelihood, a normalizing constant: f(D) = ∫ f(D|θ) f(θ) dθ over all θ
This is the version that actually matters for forecasting.
The posterior is proportional to the likelihood times the prior:
f(θ|D) ∝ L(θ) × π(θ)
(Where L(θ) denotes the likelihood and π(θ) the prior — common shorthand.)
Part V: Conjugate Priors and Analytical Solutions
In the nicest cases, the posterior has the same functional form as the prior — these are called conjugate families. They let you compute the posterior in closed form rather than numerically.
For example, if your prior on a proportion p is Beta(α, β) and you observe k successes in n Bernoulli trials, the posterior is Beta(α + k, β + n − k). Your prior mean was α/(α+β); your posterior mean is (α+k)/(α+β+n). The data literally slides your distribution along the same family.
For continuous-valued forecasts like a stock price, the normal-normal conjugacy applies: if your prior on θ is Normal(μ₀, σ₀²) and you observe data with likelihood Normal(θ, σ²), the posterior is Normal(μₙ, σₙ²) where:
μₙ = (σ² μ₀ + σ₀² x̄) / (σ² + σ₀²)
σₙ² = (σ₀² σ²) / (σ² + σ₀²)
The posterior mean is a weighted average of your prior mean and the data mean, weighted inversely by their variances. If your prior is tight (small σ₀²), the data moves you little. If your prior is diffuse, the data dominates. This is why your prior matters less as you accumulate evidence — σ₀²/(σ²+σ₀²) weights diminish as n grows.
Part VI: The Likelihood Ratio and Sequential Evidence
A more elegant way to think about updating is through the Bayes factor — the ratio of likelihoods under competing hypotheses:
K = f(D|H₁) / f(D|H₂)
The Bayes factor tells you how much the data favors H₁ over H₂, entirely independent of your prior. Your posterior odds are then:
P(H₁|D) / P(H₂|D) = K × P(H₁) / P(H₂)
Posterior odds = Bayes factor × Prior odds.
This decomposition is what makes Bayesian analysis so transparent: it separates what the evidence says (the Bayes factor) from what you believed before (the prior odds). You can disagree about priors — and you often should — but at least the evidence speaks its own truth.
For sequential data D₁, D₂, …, Dₙ under conditional independence:
K = ∏ᵢ f(Dᵢ|H₁) / f(Dᵢ|H₂)
Each new data point multiplies the Bayes factor — evidence accumulates multiplicatively. In log-space, of course, it adds:
log K = Σᵢ [log f(Dᵢ|H₁) − log f(Dᵢ|H₂)]
Part VII: Applying It to the SpaceX IPO
So what did I actually do?
I started with a prior on the IPO price. SpaceX’s last private valuation was ~$180B. Comparable public aerospace/defense multiples (Northrop Grumman, Lockheed Martin) trade at roughly 5–7× forward earnings. But SpaceX isn’t comparable — it has SpaceX (the launch business, which is high-growth), Starlink (growing satellite broadband), and the rocket-as-a-platform option value.
So my prior: a log-normal distribution centered around a $150–200B market cap, with wide tails accounting for the option value.
Then I updated on observable evidence:
- Public comparable multiples at the time of filing
- Revenue growth trajectory (Starlink profitability, launch cadence)
- Market conditions (interest rates, risk appetite)
- Pre-money investor pricing signals
Each piece of data is a likelihood function. Each updates the posterior. The posterior from one update becomes the prior for the next.
The final posterior gave me a distribution — not a single number, but a range with probabilities. The mean, median, or mode of that distribution depends on your utility function — which is a whole other conversation.
Part VIII: Why This Matters Beyond Finance
Bayes’ theorem isn’t about finance. It’s about how rational agents should think.
Every time you learn something — a news headline, a friend’s opinion, a lab result — you should update. Not by gut, not by tribal instinct, but by asking: How likely is this evidence under my current belief, and how likely is it under alternatives?
That’s it. That’s the entire rationality project compressed into one equation.
Your prior can be wrong. Your likelihoods can be misspecified. But if you do the arithmetic honestly, you converge on the truth faster than anyone who doesn’t.
And when someone asks you why you think SpaceX IPO’d at the price it did — or whether your wife is cheating — you can say:
I did the math.