50 lines
1.0 KiB
Markdown
50 lines
1.0 KiB
Markdown
---
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id: 20260114132351
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title: Bayes' Theorem
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type: permanent
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created: 2026-01-14T18:23:51Z
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modified: 2026-01-14T18:43:20Z
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tags: []
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---
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# Bayes' Theorem
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Suppose we know $P(x)$, *the prior*, and we get some data
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$Y$, and we want to know the probability $P(X|Y)$, which in
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English is *the probability of our model being correct given
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the data we've collected*.
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We can use Bayes' Theorem to find it!
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$$ P(X|Y) = \frac{P(Y|X) P(X)}{P(Y)} $$
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Where:
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- $P(X|Y)$ is called the *posterior*
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- $P(Y|X)$ is called the *likelihood*
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- $P(X)$ is called the *prior*
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- $P(Y)$ is called the *evidence*
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We can usually find the evidence by the following formula:
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$$ P(Y) = \sum P(Y|x_i) P(x_i) $$
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We can apply Bayes' Theorem to states in time too.
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$$ P(X_t|Y_t) = \frac{P(Y_t|X_t) P(X_t)}{P(Y_t)} $$
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where:
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- $X_{t+1} = f(x_t, w_t)$
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- $Y_{t} = f(x_t) + v_t$
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$w_t$ is process noise, while $v_t$ is measurement noise.
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Bayes' Rule is great for simulation and data-based
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approaches.
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## Related Ideas
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[[Expected Value]]
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## Sources
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Bayesian Signal Processing w/ Dan (1/12)
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