TITLE: Equivalent to proportionality sign for additive constants
QUESTION [11 upvotes]: Short question
Is there an equivalent to the proportionality sign $\propto$ for additive constants? The proportionality relation $y\propto x$ implies that $y=kx$ for some constant $k$. Is there a shorthand to express that $y=x+k$ for some constant $k$?
Long question
Probability distributions are often used without their explicit normalisations. E.g. for a normal distribution may be written as
$$
P(x)\propto \exp\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right).
$$
When working with log-probabilities it would be convenient to have a relation expressing that two quantities are equal up to a normalising constant such that
$$
\log P(x)\square-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2},
$$
where $\square$ is the desired relation.

REPLY [2 votes]: You can simply define a relation $\sim$ between functions such that
$$f \sim g \Longleftrightarrow \exists k \in \mathbb{R} . f = g + k$$
Or (slightly) more formally,
$$f \sim g \Longleftrightarrow \exists k \in \mathbb{R} . \forall x \in X . f(x) = g(x) + k$$
where $X$ is the domain of your functions.