关于概率密度p(x;w)与p(x|w)的差异困惑及正态分布实例咨询

Hey there, let's untangle this notation confusion—you’re definitely not the first to mix these up! The core difference lies in what the symbol after x is representing: a condition vs. a parameter.

1. p(x|w): Conditional Probability #

This is a standard conditional probability. The vertical bar | means "given that we know the value of w".

In your example, p(x|mu, sigma) = N(x|mu, sigma) translates to:

"The probability distribution of random variable x, given that we already know the values of mu and sigma, follows a normal distribution with parameters mu and sigma."

Here, mu and sigma are treated as known, fixed values that we condition on. We’re describing how x behaves when we have certainty about w.

2. p(x; w): Parametrized Probability Distribution #

The semicolon ; is used to separate the random variable (x) from the fixed parameters (w) that define the distribution.

For your example, p(x; mu, sigma) = N(x; mu, sigma) means:

"The probability distribution of random variable x is a normal distribution, which is defined by the parameters mu and sigma."

Here, mu and sigma aren’t "known conditions"—they’re just the knobs that shape the distribution. We might not know their values yet (e.g., if we’re trying to estimate them from data), but they’re not random variables themselves; they’re fixed (unknown) parameters that define how x is distributed.

Quick Rule of Thumb #

• Use | when you’re explicitly conditioning on a known value (e.g., "given that a patient has diabetes (w=1), what’s the probability of high blood sugar (x)?").
• Use ; when you’re defining a distribution that depends on parameters you might not know yet (e.g., "we model blood sugar x with a normal distribution parameterized by mu and sigma—let’s estimate mu and sigma from our patient data").

Note: Some textbooks or papers might blur these lines and use | in place of ; (or vice versa) for brevity, but the above is the standard distinction in most rigorous statistical literature.

内容的提问来源于stack exchange，提问作者baodingge