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