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Assume we know $$p(w) \sim N(0, \Sigma)$$ $$p(e) \sim N(0, 0.01)$$ $$y=w^\intercal x$$ $$o = y + e$$

Where w/x/o/y/e are all vector

How do we calculate the distribution of $$p(y|o,x)$$

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    $\begingroup$ This does not completely make sense. Assuming the "$p$" are extraneous symbols, so that we can ignore them, you seem to assert that $w$ has a multinormal distribution and $e$ has a univariate normal distribution. From the formula $y$ must be a number and so, therefore must $o$. I suspect you don't want $x$ to be a vector at all, but rather a matrix; and that you are asking for the posterior distribution of a response $y$ conditional on observing it with iid error, with a zero-mean multivariate normal prior on the regression coefficients. $\endgroup$
    – whuber
    Jan 28 '15 at 19:40
  • $\begingroup$ Sorry my bad. P means the probability of w. e is also mutinormal maybe I should make it more clear (a vector of iid) @whuber $\endgroup$ Jan 30 '15 at 18:17
  • $\begingroup$ Yes, please edit it to clear up those issues. The "$p$" notation is superfluous and potentially confusing. When $w$ is a random variable, "$w\sim N(0,\Sigma)$" means that $w$ has a $N(0,\Sigma)$ distribution. $\endgroup$
    – whuber
    Jan 30 '15 at 18:20

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