Confusing use of notation in expressing probability distribution

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SUMMARY

The discussion centers on the notation used in Bayesian statistics to express a normal distribution, specifically the expression p(x | µ, σ²). Participants highlight that this notation can be misleading, as it implies that the mean (µ) and variance (σ²) are random variables, which they are not. Instead, they serve as parameters of the Gaussian density function. The consensus is that a clearer notation should be adopted to avoid confusion between describing a probability distribution and conditional probability.

PREREQUISITES
  • Understanding of Bayesian statistics
  • Familiarity with normal distribution concepts
  • Knowledge of Gaussian density functions
  • Basic grasp of conditional probability notation
NEXT STEPS
  • Research alternative notations for expressing parameters in probability distributions
  • Explore the implications of parameterization in Bayesian models
  • Study the differences between conditional probability and probability distribution descriptions
  • Learn about common notational conventions in statistical literature
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This discussion is beneficial for statisticians, data scientists, and students of Bayesian statistics who seek clarity in the notation used for probability distributions and wish to avoid common misconceptions.

sarikan
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Hi,
I'm trying to follow a text about Bayesian statistics, and the author is using the following notation to describe a random variable which has normal distribution:

p(x | µ, σ2) = (Gaussian density function here)

In a Bayesian text, this notation is confusing, since it makes me think about mean and variance as random variables, but they are not random variables. They are simply the parameters of the density function, and this is just using the conditional probability notation for expressing something else, namely saying that "x is distributed normally, with this mean and this variance"
This is not the case where you have p(a|b) = p(b|a).p(a)/p(b)

My question is, is there a unifying way of thinking about the first notation so that I don't have to distinguish between the case where I simply have description of a probability distribution, and the case where it is about conditional distribution? I could not get my head around the idea of interpreting mean and variance as variables on which the random variable is conditioned on. Is this really a different use of the same notation, or am I missing something here?

I hope I can describe my problem, and apologies if this is not clear enough. Your response would be much appreciated!

Regards
 
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The first notation is unfortunate, since it is not a conditional probability. It would be better to use something other than | as a marker for the parameters.
 
Indeed. Especially in a Bayesian text. Thanks for the response.

Kind regards
 

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