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Properties and Applications of Binary Hypothesis Testing

Detection, Estimation, and Learning

机器学习课业代写 1 How to submit your solution • The submission deadline is 23:59 on February 4, 2022. • Submit your written responses for the questions enumerated

1 How to submit your solution

• The submission deadline is 23:59 on February 4, 2022.

• Submit your written responses for the questions enumerated below via PDF on Canvas.

• Show all the steps for obtaining results.

• The programming portion of the project is available and will be submitted via Github Classroom as described further below.

For the development and submission of your program, please use the files available via Github Classroom at https://classroom.github.com/a/IFIxZBlJ.

2 Overall Error for Binary Hypothesis Testing 机器学习课业代写

In binary hypothesis testing we consider a null hypothesis H₀ and an alternate hypothesis H₁, and an observation Y which is distributed according to p₀ : Γ → [0, ∞) if H₀ is true and p₁ : Γ → [0, ∞) if H₁ is true. We apply a test δ : Γ → {0, 1} with input Y = y to decide whether H₀ or H₁ is true. If we reject H₀ when it is true, we have a false alarm error, and we denote its  probability with P_F(δ). If we accept H0 when it is false, we have a missed detection error, and we denote its probability with P_M(δ). In the following, you will establish relationships for the overall error P_M(δ) + P_F(δ) of detection with distance measures for the underlying probability distributions.

1. [5 marks]

Show that for any α ∈ [0, 1], there is exist a test δ with P_F(δ) = α for which

P_M(δ) + P_F(δ) ≤ 1 .

2. Next, we derive relationships of the overall error for a hypothesis test with the total variance and with the Kullback-Leibler distances between p₀ and p₁.

3 Covert Communication

Consider a scenario in which Alice would like to talk to Bob without a third person, Willie, noticing the communication. This means that Alice communicates to Bob covertly, while Willie is attempting to detect an ongoing communication. (Willie will be successful just by learning of an ongoing communication. He need not be able to understand what is being said.) This can be cast into a binary hypothesis testing problem for Willie:

H₀ : Alice is not talking

H₁ : Alice is talking

4 Neyman-Pearson Test for Identifying Variable Importance 机器学习课业代写

In this last task we explore the use of the Neyman-Pearson test for selecting so-called features for a machine-learning task. Features are inputs to a machine learning model, and some features are typically more relevant for the considered classification or regression task than others. Feature selection (FS) algorithms help with identifying the more important features. The paper “A Bootstrap Based Neyman-Pearson Test for Identifying Variable Importance,” published by G. Ditzler, R. Polikar, and G. Rosen in the IEEE Transactions on Neural Networks and Learning Systems in April 2015, suggests to add another Neyman-Pearson test on top of an FS algorithm to determine if the selection of a feature by the FS algorithm is statistically relevant.

6. [3 marks]

Download the paper from the IEEE Xplore data base. If you experience problems, login via https://resources.library.ubc.ca/558/ using your UBC CWL. Read the paper in detail.

Describe in your own words how the the Neyman-Pearson test is applied in this work. Be very brief. This is only meant to gauge your understanding of this aspect of the paper.

8. [10 marks]

(a) Implement the function that performs the Neyman-Pearson test, using the python programming language.

For the development and submission of your program, please use the files available via the Github Classroom link. Use Git to clone the project directory from the Github repository created by Github Classroom. You can then navigate into the cloned folder using Terminal or Git Bash and run python main.py.

The file main.py runs two test functions, test1 and test2, which perform the first two experiments on synthetic data described in the paper. The output will be two figures that look alike Figure 1(a) and Figure 2(a) in the paper. The execution of main.py may need a few seconds. The originally produced figures are meaningless as they are based on an empty function my NP test which is not executing any Neyman-Pearson test.

You need to implement your Neyman-Pearson test in the function my NP test in the file myimplementation.py.

(b) After you implemented and tested the function, save the two figures produced in test1 and test2 and include them in your report. Briefly describe the figures and compare them to Figures 1(a) and 2(a) in the paper. To complete the assignment, use Git commands to push your changes to my NP test to Github.

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