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<metaproperty="twitter:description" content="Interactive mathematical equations with KaTeX rendering. Write equations in LaTeX and descriptions in Markdown, with color-coded terms that reveal their meaning on hover. Built with TypeScript." />
Your [updated belief]{.posterior} is your [prior view]{.prior} re-weighted by its [predictive power]{.likelihood} and normalized by the [total probability of the evidence]{.evidence}.
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## .posterior
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Posterior Probability $P(H|E)$.
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What you believe *after* seeing the data. It is the probability of the Hypothesis ($H$) being true given the Evidence ($E$). This is the output of the learning process.
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## .likelihood
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Likelihood $P(E|H)$.
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How well the hypothesis explains the data. It asks: "If my theory were true, how likely would this outcome be?" High likelihood means the theory strongly predicts the observed evidence.
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## .prior
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Prior Probability $P(H)$.
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Your starting assumption *before* seeing new data. It represents base rates or previous knowledge. Strong priors are hard to shift; weak priors change easily with new evidence.
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## .evidence
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Marginal Likelihood $P(E)$.
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The total probability of seeing the evidence under *all* possible hypotheses. It acts as a normalization factor. If the evidence is surprising (low $P(E)$), it has a stronger effect on updating your beliefs.
To find [the amplitude]{.amplitude} [at a particular frequency]{.freq}, [spin]{.spin} [your signal]{.signal} [around a circle]{.circle} [at that frequency]{.freq}, and [average a bunch of points along that path]{.average}.
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## .amplitude
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The transform output $X_k$.
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Represents the **complex amplitude** (magnitude and phase) of the signal at a specific frequency. Its magnitude tells you "how much" of that frequency is present.
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## .freq
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The frequency index $k$.
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Determines which frequency we are analyzing. It appears both in the output index (which bin?) and in the rotation term (how fast do we spin?).
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## .average
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The averaging operation.
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Combines the summation ($\sum$) and the division by $N$ (and the time index term $\frac{n}{N}$). It turns the sum into an average value over the signal's duration.
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## .signal
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The input signal $x_n$.
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Your raw data points (audio, image, stock prices) sampled over time.
The [Ricci curvature]{.ricci} and [scalar curvature]{.scalar} combined with [dark energy]{.dark-energy} are determined by the [matter distribution]{.matter} scaled by the [gravitational constant]{.coupling}.
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## .ricci
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Ricci Curvature Tensor ($R_{\mu\nu}$).
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This describes how the volume of a shape changes as it moves through spacetime. It represents the part of curvature that causes matter to converge or diverge, directly corresponding to the presence of mass and energy.
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## .scalar
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Scalar Curvature Term ($-\frac{1}{2}R g_{\mu\nu}$).
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This geometric correction involves the Ricci scalar $R$ and metric tensor $g_{\mu\nu}$. It ensures that the geometry side of the equation mathematically conserves energy and momentum, matching the physics of the matter side.
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## .dark-energy
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The Cosmological Constant ($\Lambda$).
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An intrinsic energy of empty space that pushes the universe apart. It opposes gravity and is responsible for the accelerating expansion of the universe.
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## .coupling
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Gravitational Coupling.
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This tiny factor ($\approx 2 \times 10^{-43}$) describes how "stiff" spacetime is. It explains why gravity is so weak: it takes a massive amount of matter (like the Earth) to produce even a modest curvature.
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## .matter
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Stress-Energy Tensor ($T_{\mu\nu}$).
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The source of gravity. It tells space how to curve using mass density ($\rho$), pressure ($p$), and momentum flux. In General Relativity, all energy gravitates, not just mass.
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