If we were learning or working in the machine learning field, then we frequently come across the term "probability distribution". I know what probability, conditional probability, and probability distribution/density in math mean, but what is its meaning in machine learning?

Take this example where $x$ is an element of $D$, which is a dataset,

$$x \in D$$

Let's say our dataset ($D$) is MNIST with about 70,000 images, so then $x$ becomes any image of those 70,000 images.

In many papers and web articles, these terms are often denoted as probability distributions

$$p(x)$$

or

$$p\left(\frac{z}{x} \right)$$

• What does $p(\cdot)$ even mean, and what kind of output does $p(\cdot)$ give?
• Is the output of $p(\cdot)$ a scalar, vector, or matrix?
• If the output is vector or matrix, then will the sum of all elements of this vector/matrix always be $1$?

This is my understanding,

$p(\cdot)$ is a function which maps the real distribution of the whole dataset $D$. Then $p(x)$ gives a scalar probability value given $x$, which is calculated from real distribution $p(\cdot)$. Similar to $p(H)=0.5$ in a coin toss experiment $D={\{H,T}\}$.

$p\left(\frac{z}{x} \right)$ is an another function that maps the real distribution of the whole dataset to a vector $z$ given an input $x$ and the $z$ vector is a probability distribution that sums to $1$.

Are my assumptions correct?

An example would be a VAE's data generation process, which is represented in this equation

$$p_\theta(\mathbf{x}^{(i)}) = \int p_\theta(\mathbf{x}^{(i)}\vert\mathbf{z}) p_\theta(\mathbf{z}) d\mathbf{z}$$

If we were learning or working in the machine learning field, then we frequently come across the term "probability distribution". I know what probability, conditional probability, and probability distribution/density in math mean, but what is its meaning in machine learning?

Take this example where $x$ is an element of $D$, which is a dataset,

$$x \in D$$

Let's say our dataset ($D$) is MNIST with about 70,000 images, so then $x$ becomes any image of those 70,000 images.

In many papers and web articles, these terms are often denoted as probability distributions

$$p(x)$$

or

$$p\left(z \mid x \right)$$

• What does $p(\cdot)$ even mean, and what kind of output does $p(\cdot)$ give?
• Is the output of $p(\cdot)$ a scalar, vector, or matrix?
• If the output is vector or matrix, then will the sum of all elements of this vector/matrix always be $1$?

This is my understanding,

$p(\cdot)$ is a function which maps the real distribution of the whole dataset $D$. Then $p(x)$ gives a scalar probability value given $x$, which is calculated from real distribution $p(\cdot)$. Similar to $p(H)=0.5$ in a coin toss experiment $D={\{H,T}\}$.

$p\left(z \mid x \right)$ is another function that maps the real distribution of the whole dataset to a vector $z$ given an input $x$ and the $z$ vector is a probability distribution that sums to $1$.

Are my assumptions correct?

An example would be a VAE's data generation process, which is represented in this equation

$$p_\theta(\mathbf{x}^{(i)}) = \int p_\theta(\mathbf{x}^{(i)}\vert\mathbf{z}) p_\theta(\mathbf{z}) d\mathbf{z}$$

If we were learning or working in machine learning field then we frequently come across this term probability distribution. I know what probability, conditional probability and probability distribution/density in math means but what is its meaning in machine learning?

Take this example where x is an element of D, D begin a dataset,

$x \in D$

Let say our dataset(D) is MNIST with about 70,000 images,so then x becomes any image of those 70,000 images.

In many papers and web articles we notice these terms as probabiltiy distributions

$p(x),\ p(\frac{z}{x})$

• What is p() even means and what kind of output does p() gives?
• Is the output of p() scalar, vector or matrix?
• if the output is vector or matrix then will the sum of all elements of this vector/matrix always be 1?

This is my understanding,

p() is a function which maps the real distribution of the whole dataset D. Then $p(x)$ gives a scalar probability value given x which is calculated from real distribution p(). Similar to p(H)=0.5 in a coin toss experiment $D={\{H,T}\}$.

$p(\frac{z}{x})$ is an another function which maps the real distribution of the whole dataset to a vector z given an input x and the z vector is a probability distribution which sums to 1.

Are my assumptions correct?

EDIT:

An example would be a VAE's data generation process which is represented in this equation $p_\theta(\mathbf{x}^{(i)}) = \int p_\theta(\mathbf{x}^{(i)}\vert\mathbf{z}) p_\theta(\mathbf{z}) d\mathbf{z}$

If we were learning or working in the machine learning field, then we frequently come across the term "probability distribution". I know what probability, conditional probability, and probability distribution/density in math mean, but what is its meaning in machine learning?

Take this example where $x$ is an element of $D$, which is a dataset,

$$x \in D$$

Let's say our dataset  ($D$) is MNIST with about 70,000 images, so then $x$ becomes any image of those 70,000 images.

In many papers and web articles, these terms are often denoted as probability distributions

$$p(x)$$

or

$$p\left(\frac{z}{x} \right)$$

• What does $p(\cdot)$ even mean, and what kind of output does $p(\cdot)$ give?
• Is the output of $p(\cdot)$ a scalar, vector, or matrix?
• If the output is vector or matrix, then will the sum of all elements of this vector/matrix always be $1$?

This is my understanding,

$p(\cdot)$ is a function which maps the real distribution of the whole dataset $D$. Then $p(x)$ gives a scalar probability value given $x$, which is calculated from real distribution $p(\cdot)$. Similar to $p(H)=0.5$ in a coin toss experiment $D={\{H,T}\}$.

$p\left(\frac{z}{x} \right)$ is an another function that maps the real distribution of the whole dataset to a vector $z$ given an input $x$ and the $z$ vector is a probability distribution that sums to $1$.

Are my assumptions correct?

An example would be a VAE's data generation process, which is represented in this equation

$$p_\theta(\mathbf{x}^{(i)}) = \int p_\theta(\mathbf{x}^{(i)}\vert\mathbf{z}) p_\theta(\mathbf{z}) d\mathbf{z}$$