Understanding Empirical Data Distribution"We can approximate p ( x , y ) = p ( x ) p ( y | x ) using the empirical data distribution p ( x , y ) = 1 N ∑ n = 1 N δ x n ( x ) δ y n ( y ) "In another part of the paper they say p ( y | y n ) = δ y n ( y ).I have some background in probability but none in statistics; I was able to figure out what an Empirical CDF is, but not a pdf like here, so I'm not sure exactly what the authors are doing. Does the δ refer to the Dirac delta distribution?

Question

Understanding Empirical Data Distribution&quot;We can approximate   p  (  x  ,  y  )  =  p  (  x  )  p  (  y      |    x  ) using the empirical data distribution  p  (  x  ,  y  )  =      1    N        ∑          n      =      1        N        δ                  x        n              (  x  )      δ                  y        n              (  y  )  &quot;In another part of the paper they say   p  (  y      |        y    n    )  =      δ                  y        n              (  y  ).I have some background in probability but none in statistics; I was able to figure out what an Empirical CDF is, but not a pdf like here, so I&#039;m not sure exactly what the authors are doing. Does the   δ refer to the Dirac delta distribution?

Ryan Davies · Accepted Answer

Step 1The empirical data distribution is a probability distribution which allocates probability 1/N to point in the training dataset and 0 otherwise. More formally, it is supported on N points   (      x    i    ,      y    i    ) of the training set each having probability mass 1/N$ and so all other points have mass 0.Step 2Yes,       δ                  x        n              (  x  ) and       δ                  y        n              (  y  ) are indicator functions which are 1 when   x  =      x    n   and 0 otherwise; similarly for       δ                  y        n              (  y  ).P(x,y) is the joint probability mass and you can check that it is 0 for any point not in the training set and 1/N for a point in the training set.

"We can approximate p(x,y)=p(x)p(y|x) using the empirical data distribution p(x,y) =1/N sum_{n=1}^N delta_{x_n}(x) delta_{y_n}(y)"

Answered question

Answer & Explanation

New Questions in High school statistics