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 ) ,,whereX 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?

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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  )  ,,whereX 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&#039;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?

Ullveruxqte · Accepted Answer

I think you lack relevant knowledge on Riemann–Stieltjes integralIn 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)dxI hope it is useful to you. Sorry, if it didn&#039;t help you.

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

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