I am quoting this from DeGroot's "Probability and Statistics".
Let $X$ and $Y$ be random variables. The joint probability density function(p.d.f.) $f$ defines a surface over the $xy$-plane for which the height
$f(x, y)$ at each point $(x, y)$ represents the relative likelihood of that point. For
instance, if it is known that $Y = y_0$, then the point $(x, y)$ must lie on the line $y = y_0$ in
the $xy$-plane, and the relative likelihood of any point $(x, y_0)$ on this line is $f(x, y_0)$.
I don't know the meaning of "relative likelihood".Please explain.
Best Answer
I think the quote is using the term "likelihood" rather loosely. Likelihood is a separate concept from probability. What the joint density function gives at each point is a....density (hence the name). I don't know why the text said relative likelihood, which is $L_{relative}(a,b)=\frac{L(a)}{L(b)}$, where the likelihood function is evaluated given a set of data with the parameters becomming the variables: i.e., L(a|X) = P(X|a), where X is a set of data or outcomes and a is a parameter of the distribution.
The quote above appears to be talking about the conditonal probability of x given $y_{0}$, which is $p_{X|Y}(x|y_{0}) = p_{XY}(y_{0},x)/\int_x p_{XY}(y_{0},x) dx$, in that case the joint density evaluated at $y_{0}$ is acting in a similar manner to the likelihood function in Bayesian inference, but the two are rather different concepts.
Hope that helps.