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What do the vertical bars mean in the first and third formulae?

$$v_i|z_i=k,\mu_k\sim\mathcal{N}(\mu_k, \sigma^2)$$ $$P(z_i=k)=\pi_k$$ $$\pi|\alpha\sim \text{Dir}(\alpha/K1_K)$$ $$\mu_k\sim H(\lambda)$$ This formula is originally from here.

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I think in this context the vertical bars can read as "given that". So the first line would mean $v_{i}$ given that $z_{i} = k,\mu_{k}\sim N(\mu_{k},\sigma^{2})$. See also this list on Wikipedia. –  COOLSerdash Jul 31 at 20:28

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The vertical bar is often called a 'pipe'. It is often used in mathematics, logic and statistics. It typically is read as 'given that'. In probability and statistics it often indicates conditional probability, but can also indicate a conditional distribution. You can read it as 'conditional on'.

For example the third line can be read "pi, conditional on alpha, is distributed as dirichlet... ". The idea of a distribution conditional on something else taking a specific value is very, very common in statistics. Perhaps the most typical example would be of $Y$ values conditional on $X$ being normally distributed in regression models (for an example, see my answer here: What is the intuition behind conditional Gaussian distributions).

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I know the idea of conditional probability, what confused me here is that, usually the conditional probability is written as P(A|B), which is different with this case. And how to interpret the first formula? –  Jiang Xiang Jul 31 at 20:43
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"The probability of $v_i$, given that $z_i = k$ and given the value $\mu_k$, is distributed normally with mean $\mu_k$ and variance $\sigma^2$." –  user777 Jul 31 at 20:48
    
Sorry about that, @JiangXiang. In doesn't have to be conditional probability, it can just indicate a conditional distribution, eg. I edited to clarify that. I didn't mention the 1st formula, because COOLSerdash had already discussed it. I also agree w/ user777's reading (they are the same). –  gung Jul 31 at 20:53
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It would clarify matters to define what you mean by a "conditional distribution" and explicitly distinguish it from a "conditional probability." The uses of "$|$" in the first and third lines are really quite different, despite the similarity of notation, so relying on the very same word "conditional" to distinguish them might be more confusing that enlightening. –  whuber Jul 31 at 21:04
    
Thanks. For the particular problem in Dirichlet Process, in the first formula $$v_i|z_i=k,\mu_k\sim\mathcal{N}(\mu_k, \sigma^2)$$ it would be safe to ignore the $\mu_k$: $$v_i|z_i=k\sim\mathcal{N}(\mu_k, \sigma^2)$$ because as long as $z_{i}$belongs to the k-th cluster, its mean should be $\mu_k$. Is it necessary to put the $\mu_k$ here? Or there are some other reasons for this? –  Jiang Xiang Jul 31 at 21:07

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