On integration over distribution. When I read various journal articles related to machine learning,

Sufirk6u38

Sufirk6u38

Answered question

2022-06-01

On integration over distribution.
When I read various journal articles related to machine learning, I often face integrals over distribution.
In an article I am reading now, for example, a risk function associated with distribution ϕ is defined by
R i ( θ ) = f L ( f θ ( x ) , y ) d ϕ ( x , y ) ,,
where
X and Y are a feature space and a label space, respectively.
f θ : X Y is a given model parameterized by θ Θ,
f L : X × Y R 0 is a loss function, and
ϕ is the data generating distributions.
In addition to the above case and the others (even not related to this field), I have seen many times integration formulas over distributions. However, whenever I encountered them, I couldn't grasp what it is.
Rather, I am familiar with the following equation:
f L ( x ) p X ( x ) d x ,
where fL(x) is a cost (or reward) function achieved by an event x, and pX(x) is a probability that an event x occurred.
Can someone please let me know what the integral over a distribution means?

Answer & Explanation

Ullveruxqte

Ullveruxqte

Beginner2022-06-02Added 4 answers

I think you lack relevant knowledge on Riemann–Stieltjes integral
In the Probability theory,the Expectation of Discrete distribution and continuous distribution can be written uniformly as Riemann–Stieltjes integral form, that is
E X ( f ( x ) ) = f ( x ) d F ( x )
where F(x) is the cumulative distribution function of random variable X, you can roughly think that dF(x)=p(x)dx
I hope it is useful to you. Sorry, if it didn't help you.

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