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.

Solution 1:

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.