# Notation: in general is it ok to think of $p(y|x)$ as $p_x(y)$?

In the notation for the conditional probability, $$p(y|x)$$, I believe usually we think of this with $$x$$ having a fixed value and $$y$$ varying, such that it integrates to one.

In this case, is it acceptable to instead write it, or think of it, as $$p_x(y)$$?

I.e. $$p_x(y) \equiv p(y|x)$$

Somehow I would find this to be more clear than the traditional conditional notation.

## 2 Answers

This is not the best choice for notation. Such notation is commonly used in two scenarios.

First, you can see people writing $$p_X(x)$$, where they mean "value returned by probability distribution $$p$$ of random variable $$X$$ for $$x$$". In this case $$p(x)$$ is a less precise shorthand for $$p_X(x)$$. So if $$X$$ is random variable for heights of males and $$Y$$ is heights of females, then $$p_X(y)$$ means "what would be the probability of observing female's height $$y$$ according to the distribution of male heights $$p_X$$". Using this notation, people write $$p_{X|Y}(x)$$ to denote $$p(X=x|Y)$$. You can find many uses of this notation in Wikipedia's article on conditional probability distributions.

Second common usage, is writing $$f_\theta(x)$$ for "probability density (or probability) returned by probability density (or mass) function $$f$$ parametrized by $$\theta$$ for $$x$$". In this case again $$\theta$$ is a property of $$f$$. So $$f_\theta(x)$$ is a different spelling of $$f(x; \theta)$$.

Moreover, you write

In the notation for the conditional probability, $$p(y|x)$$, I believe usually we think of this with $$x$$ having a fixed value and $$y$$ varying [...]

but notice that $$P(X=x|Y=y)$$ and $$P(X=x|Y)$$ are not the same, since in first case you ask about conditioning on fixed value of $$Y$$, while in the second case you ask about conditioning on random variable.

- There is nothing special about letter $$f$$, you can use any other letter in here, $$f$$ and $$p$$ are the two most popular choices.

You can use if you’re consistent in your notation (but I don't really suggest). Although widely used, $$p(x), p(y), p(x|y),...$$ are all abuse of notations. They’re ambigious, e.g. what is $$p(2)$$? $$p$$ doesn’t represent the pdf of a specific RV.

A better and fully unambigious way is to use $$p_X(x)$$, where subscript denotes the RV, and inside parantheses the specific value. This way we know which pdf we use.

Your new notation actually may cause people to think “put $$y$$ in the pdf of $$x$$”. If you’ll only read your notes, it’s OK upto some extent.

A bad case when you don’t use subscripts is the convolution of densities by the way: $$\int f_X(x)f_Y(z-x)dx \ \ \text{vs} \ \ \int f(x)f(z-x)dx$$

In the RHS expression, it's totally ambiguous what $$f(.)$$ represents.