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hypothesis testing

1 terminology

  • \(p_{\mathsf{H}}(H_m)\) is the a priori probability of hypothesis \(H_m\)
  • \(p_{\mathsf{y}\mid\mathsf{H}}(\cdot \mid H_m)\) is the conditional probability of the observed data under \(H_m\)
    • Can think of as "if \(H_m\) is true, then this is what the distribution would look like"
    • Or can think of as "\(H_1, .., H_2\) and \(\mathbf{y}\) are random variables that label each outcome \(\omega\) in the probability space. \(p(\mathbf{y} = y \mid H_m = h)\) is the probability of all the \(\omega\) that satisfy \(\mathbf{y}(\omega) = y\) and \(H_m(\omega) = h\), normalized by all the probability mass that \(H_1\) takes up.
  • \(p_{\mathsf{H} \mid \mathsf{y}} (H_m \mid \mathbf{y})\) is the a posteriori probability of a hypothesis given an observation

2 TODO binary hypothesis testing

3 decision rule

The solution to a hypothesis testing problem is a decision rule.

See Bayesian Inference note for more discussion.

4 see also

5 sources

Created: 2021-09-14 Tue 21:44