Beta distribution is conjugate to binomial likelihood
Beta-binomial distribution
Beta distribution is conjugate to binomial likelihood
The beta distribution serves as a conjugate prior for the binomial distribution in Bayesian statistics. This conjugacy simplifies the process of updating beliefs with new data, as the posterior distribution remains within the same family of distributions.
Example
Suppose we have 10 coin flips with an unknown probability of heads. If we start with a beta distribution prior for the probability of heads, after observing the outcomes, our posterior distribution will also be a beta distribution, reflecting the updated beliefs about the probability of heads.
Understanding conjugate priors like the beta distribution is crucial for efficient Bayesian inference, as it allows for straightforward updating of probability distributions with new evidence.
Related concepts
Conjugate prior
Conjugate priors simplify Bayesian updating
the β₁ and β₂ hyperparameters control in Adam
β₁ controls the exponential decay rate of the first moment estimates; β₂ controls the exponential decay rate of the second moment estimates in Adam optimizer
Resampling (statistics)
Bootstrapping samples with replacement to estimate distributions
importance sampling does: reweights samples from proposal to estimate target expectation
Importance sampling reweights samples from a proposal distribution to estimate the expectation under a target distribution
Nyquist–Shannon sampling theorem
Sample at ≥ 2× the highest frequency to avoid aliasing
the Dirichlet distribution does: distribution over probability simplices
The Dirichlet distribution generates random probability vectors over a simplex
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