Difference Between Semicolon and Vertical Bar in Probability

Summary:

• $ P(x; \theta) $: A parameterized probability distribution with parameters $ \theta $.
  • $ x $ means a random variable, $ \theta $ represents the parameters of distribution.
  • Example: In a normal distribution $ P(x; \mu, \sigma) $, parameters $ \mu $ and $ \sigma $ respectively represent mean and standard deviation.
• $ P(x \mid \theta) $: The conditional probability of $ x $ given events $ \theta $ has occurred or is known.
  • Example: In Bayesian statistics, a likelihood function $ P(x; \theta) $ represents the probability of the observed data $ x $ given the parameter $ \theta $.

References:

• Microsoft Copilot
• https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols
• https://en.wikipedia.org/wiki/Conditional_probability
• https://stats.stackexchange.com/questions/10234/
• https://stats.stackexchange.com/questions/301382/
• https://blog.neko-ni-naritai.com/entry/diff-of-semicolon-and-pipe