In regression, what is the likelihood Notation for regression,
$$ 
  y = f_w(x)
$$
The regression function f() takes $x$ as input and is parameterized by $w$.
The posterior is $p(w|x,y)$.
Is the likelihood is then $p(x,y|w)$?   
Or is it $p(y|x,w)$.  
The first choice $p(x,y|w)$ would seem correct in that we have just switched the order of things across the conditional.
On the other hand, it also feels wrong, in that the regression input $x$ does not depend on $w$,
whereas $y$ does depend on $x$.
 A: I think you mess together two things here. For the purpose of inference of weights of your regression model, you want to compute the posterior $p(w|x,y)$ which is proportional to the product of the likelihood and the prior, $p(x,y|w)p(w)$. In this context, the likelihood refers to your first proposal and evaluates how likely is a pair $(x,y)$ to appear according to your model with weights set to some particular $w$.
The other thing you talk about is the predictive distribution: given a new sample $x'$, what is the distribution of $f(x')=y'$ according to the model? This is denoted $p(y'|x',x,y)$: it depends on the current input $x'$ and the training data $x, y$. It can be further expanded to $\int p(y'|x',w)p(w|x,y)\,\mathrm dw$.
This video provides some more explanation on the matter.
A: If I have read the question correctly, then you are right that $x$ does not depend on $w$. $x$ is the data that we are trying to learn from. 
The likelihood is the probability of observing a given set of observations, given the value of the parameters, that is $p(x,y|w)$
