Consider histogram-based density estimation for some PDF p(x) defined on the interval [0, 1]. Suppose that...

 

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Consider histogram-based density estimation for some PDF p(x) defined on the interval [0, 1]. Suppose that we are given a training set D of n samples drawn from p(x), D = {X1, X2, ..., xn}. Further suppose we use the following m equal-length bins for computing the histogram: B₁=[0, 1/m), B2=[1/m, 2/m), Bm=[(m-1)/m, 1]. With this, we may count the number of samples falling into each bin, and we denote that number by Y; for the j-th bin. (a) Write down the histogram-based density estimate p(x), which should be a function of x and those quantities given above. Note: you need to write down a close-formed estimate so that you may evaluate its value for any x. (b) For a given x, find the expectation of your estimate p(x), i.e., E[ô(x)]. Consider histogram-based density estimation for some PDF p(x) defined on the interval [0, 1]. Suppose that we are given a training set D of n samples drawn from p(x), D = {X1, X2, ..., xn}. Further suppose we use the following m equal-length bins for computing the histogram: B₁=[0, 1/m), B2=[1/m, 2/m), Bm=[(m-1)/m, 1]. With this, we may count the number of samples falling into each bin, and we denote that number by Y; for the j-th bin. (a) Write down the histogram-based density estimate p(x), which should be a function of x and those quantities given above. Note: you need to write down a close-formed estimate so that you may evaluate its value for any x. (b) For a given x, find the expectation of your estimate p(x), i.e., E[ô(x)].