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From my understanding the posterior "over" parameters $\alpha$ is

$$p(D|\alpha)$$

and not

$$p(\alpha|D),$$

is it correct?

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    $\begingroup$ Given different people might use the same phrase to mean somewhat different things, some context and definitions might help. Presumably $\alpha$ is a parameter vector, but does it include all the parameters? What's $D$? Who is using the term posterior over parameters, and what else are they saying near it? $\endgroup$ – Glen_b Jan 11 '15 at 0:05
  • $\begingroup$ In the example of the question, with $D$ I generically meant "data", and $\alpha$ a vector of parameters... (what do you mean with "all" the parameters, and why it matters?). Here just two examples that seem to contradict my understanding. Look for "posterior over":robots.ox.ac.uk/~parg/projects/ica/riz/Thesis/thesis019.html astro.uni-bonn.de/~kbasu/ObsCosmo/Slides2013/obs_cos_4_2013.pdf $\endgroup$ – njk Jan 11 '15 at 0:22
  • $\begingroup$ If there are several parameters, for example $\mu, \sigma$, and $\kappa$ (say), and $\alpha=(\sigma,\kappa)$, then there's an additional possible meaning of the phrase -- $p(\mu|D,\alpha)$. $\endgroup$ – Glen_b Jan 11 '15 at 0:26
  • $\begingroup$ If the options in your question are the only two possible, I could only assume the second is the intent. The first is a likelihood, not a posterior. $\endgroup$ – Glen_b Jan 11 '15 at 0:28
  • $\begingroup$ Note that in both your example links, there are in fact additional parameters the first time the phrase is used (hyperparameters included in $M$ in the first, and everything but $\theta_2$ in the second). [Generally speaking, something like $p(\alpha|D,M)$ will be nearer the intent, but sometimes "over" might be used to mean conditioning on $\alpha$, as in $p(\theta|\alpha,D,M)$; but if that happens, it's almost always obvious from context] $\endgroup$ – Glen_b Jan 11 '15 at 0:33
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No.

The standard convention is that posterior [distribution] over parameter $\alpha$ is

$$ P(\alpha | D) $$

Just like $P(x)$ is some distribution over $x$, $P(\alpha | D)$ is a distribution over $\alpha$.

Posterior here means that we condition $\alpha$ on given data $D$. If we didn't, it'd be a prior (in Bayesian terms) [distribution] $P(\alpha)$.

$P(D|\alpha)$ is not a distribution over $\alpha$, it's called likelihood.

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