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Related question: What is the meaning of pyx theta and px theta and px yz

In this question, doesn't the p represent the density function? Then why is it written in a lot of text as probability? For example, $p(y^i|x^i;\theta)$ is written as probability of $y^i$ given $x^i$ parametrized by theta, though it should be probability density of $y^i$ given $x^i$ parametrized by $\theta$.

As asked in comment -

Djib2011
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levio_sa
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    Welcome to CV, levio_sa. You should edit your question ("edit" link at bottom left of it) to erase the "Unrelated to the above question…" and subsequent material (and put them in two **new questions**! :) – Alexis Sep 05 '21 at 16:01
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    Not every text uses $p$ exclusively for denoting probability. It can also denote a density, as you've found. Notation conventions will be different for every text, so it's important to pay attention to how an author defines a symbol. The only place that these symbols are described in the linked question is as "probabilities," so it's not even the case that there is a contradiction in that resource; the author does not describe densities at all! – Sycorax Sep 05 '21 at 16:07
  • A text I am using says The notation $p(y^i|x^i; θ)$ indicates that this is the distribution of $y^i$ given $x^i$ and parameterized by θ. After a few lines it says The probability of the data is given by $p(\overrightarrow{y}|X; θ)$. I am unable to get why they have used two distribution and probability but not density. – levio_sa Sep 05 '21 at 16:10
  • Also, even in probability density, the input is a value not a random variable. For example, p(Y=y|X=x) = some function of y which also depends on x. – levio_sa Sep 05 '21 at 16:17
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    it will be easier to understand and answer your question if post a quotation from the text that you want to know about instead of linking to an unrelated question – Sycorax Sep 05 '21 at 16:22
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    Hi, there are blind and visually impaired users of this site who interact with it using screen readers. The screen readers can't handle the equation in your screenshot. Please edit the post to include the equation as LaTeX. If it helps, we have some [resources on using LaTeX on Cross Validated](https://stats.meta.stackexchange.com/a/1605/155836). – kjetil b halvorsen Sep 06 '21 at 02:40
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    There are fashions in terminology as well as variations in rigour over time and between authors. The terminology of _probability density function_ was introduced to make a distinction more evident. Then the term _probability mass function_ was introduced to make it even more emphatic. But that didn't settle usage for eternity. Some texts have rowed back and insisted that probability density is a general idea; it's just a matter of what measure is implied. It could be counting measure. Usage depends on the taste of authors and their expectations of how much mathematics readers should know. – Nick Cox Sep 06 '21 at 12:02

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It’s not. As already said in the comments $p$ is not reserved in mathematics for a probability, as any other letter or symbol isn’t. The meaning would follow from context. It is often used for probability density as in your example.

The quote says that $p$ stands for “distribution”, that’s imprecise. Distribution can be described by a probability mass function (discrete), probability density function (continuous), or cumulative distribution function. Later they use the word “probability” but they do it loosely. Obviously, the discussed function is a Gaussian probability density function, so it’s rather probability densities. It’s imprecise, but you would see people often saying “probability” when they mean “probability density”.

Tim
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