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(e) Assuming a uniform distribution over all bins, what is the prob- ability that a particular bin is empty, as a function of d, m and n?

(e) Assuming a uniform distribution over all bins, what is the prob- ability that a particular bin is empty, as a function of d, m and n?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Consider a dataset {r;}"-1 where each point r E [0, 1]d. Let f(x) be the true unknown data distribution. You decide to use a histogram method to estimate the density f(x) and divide each dimension into m bins. (a) Show that for a measurable set S, E[1{resy] = P(r E S). (b) Combining the result of the previous question with the Law of Large Numbers, show that the estimated probability of falling in bin i, as given by the histogram method, tends to fv. f(x)dx, the true probability of falling in bin i, as n → o. V; denotes the volume occupied by bin i. 4
Transcribed Image Text:Consider a dataset {r;}"-1 where each point r E [0, 1]d. Let f(x) be the true unknown data distribution. You decide to use a histogram method to estimate the density f(x) and divide each dimension into m bins. (a) Show that for a measurable set S, E[1{resy] = P(r E S). (b) Combining the result of the previous question with the Law of Large Numbers, show that the estimated probability of falling in bin i, as given by the histogram method, tends to fv. f(x)dx, the true probability of falling in bin i, as n → o. V; denotes the volume occupied by bin i. 4
(c) Consider the MNIST dataset with 784 dimensions. We divide each dimension into 2 bins. How many digits (base 10) does the total number of bins have? (d) Assume the existence of an idealized MNIST classifier based on a histogram of m = 2 bins per dimension. The accuracy of the classifier increases by e = 5% (starting from 10% and up to a maximum of 100%) each time k = 4 new data points are added to every bin. What is the smallest number of samples the classifier requires to reach an accuracy of 90%? (e) Assuming a uniform distribution over all bins, what is the prob- ability that a particular bin is empty, as a function of d, m and n? Note the contrast between (b) and (e): even if for infinitely many datapoints, the histogram will be arbitrarily accurate at estimating the true distribution (b), in practice the number of samples required to even get a single datapoint in every region grows exponentially with d.
Transcribed Image Text:(c) Consider the MNIST dataset with 784 dimensions. We divide each dimension into 2 bins. How many digits (base 10) does the total number of bins have? (d) Assume the existence of an idealized MNIST classifier based on a histogram of m = 2 bins per dimension. The accuracy of the classifier increases by e = 5% (starting from 10% and up to a maximum of 100%) each time k = 4 new data points are added to every bin. What is the smallest number of samples the classifier requires to reach an accuracy of 90%? (e) Assuming a uniform distribution over all bins, what is the prob- ability that a particular bin is empty, as a function of d, m and n? Note the contrast between (b) and (e): even if for infinitely many datapoints, the histogram will be arbitrarily accurate at estimating the true distribution (b), in practice the number of samples required to even get a single datapoint in every region grows exponentially with d.
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