Prior
A belief before evidence.
P(A)
∫₀¹ f(x)dx = 1
kakuritsu
a theater for probability — the quiet architecture of likelihood
P(A∩B) = P(A)·P(B|A)
§ 01 — Distribution Canvas
The shape of expectation
A grid governed by a Gaussian curve. The weight of each card follows a density function — most mass at the center, attenuating to the margins.
Conditionality
The probability of one event given the presence of another — knowledge as constraint.
P(A|B)
The mean of belief
Every distribution has a center. Here the expected value rests in full view, the axis around which uncertainty is measured. Beneath it, variance organizes itself into symmetric descent.
E[X] = Σ xᵢ · p(xᵢ)
Likelihood
A function of the data, measured against a hypothesis.
L(θ|x)
Tail
Rare outcome.
P(|X|>3σ)
Bayes' theorem
A rule for updating belief in the light of new evidence.
P(A|B) = P(B|A)·P(A) / P(B)
Convergence of sums
Independent draws from any distribution, averaged, tend toward normality. The bell curve emerges not by decree but by accumulation — a theorem that keeps reasserting itself in the presence of enough samples.
(X̄ − μ) · √n / σ → N(0, 1)
Second moment
The expected squared deviation from the mean.
σ² = E[(X − μ)²]
Maximum entropy
When nothing is known, every outcome is equally admissible.
H(X) = −Σ p(x) log p(x)
The null hypothesis
A default assumption of no effect. Rejected only when the evidence is sufficiently improbable under its terms.
H₀ : θ = θ₀
Belief after evidence
The updated distribution, once the data has spoken.
p(θ|x) ∝ p(x|θ) · p(θ)
lim n→∞
§ 02 — Convergence Strip
The law of large numbers
As the sample grows, the sample mean draws nearer to the expected value. Variation is not abolished; it is merely averaged away. What seems chaotic in a single trial resolves, under sufficient repetition, into the lawful shape of its own expectation. The many become the one — not by erasure, but by arithmetic.
X̄ₙ → μ (as n → ∞)
X ~ U(0,1)
§ 03 — Scatter Field
Sampling the unordered
A moment of released constraint. Each card is a realized value — dropped from a distribution, finding its own coordinate in the probability plane.
Uniform
Every outcome equally admissible within its range.
X ~ U(a, b)
Poisson
Counts of rare events arriving at a steady average rate.
P(k) = λᵏe⁻ᵏ/k!
Exponential
The interval between consecutive memoryless events.
f(x) = λe⁻ˣλ
Binomial
The sum of independent Bernoulli trials — success is counted.
P(k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ
Gaussian
The limit shape of aggregated independence.
N(μ, σ²)
Geometric
Number of trials before a first success.
P(k) = (1−p)ᵏ⁻¹ p
Beta
A distribution over probabilities themselves.
Beta(α, β)
Bernoulli
A single binary trial — the atom of probability.
P(X=1) = p
Log-normal
The multiplicative cousin of the Gaussian.
ln X ~ N(μ, σ²)
A single point, certified.
Every probability ends where it began — a point in the plane, a claim awaiting evidence. The theater closes not with conclusion but with continuation.
P = 1.00