## Description

**Mathematics:**- Consider a Bayesian classification problem where we wish to determine if a recentlyregistered user of our Fordham network is a student, a faculty member, or a dangerous hacker. We use four features – x
_{1 }(VisitFrequency), x_{2 }(LoginLocation), x_{3}(LoginDuration), x_{4}SoftwareUsed . Each feature takes on one of five values, shown below:

VisitFrequency | Never | Monthly | Daily | ||

LoginLocation | OnCampus | InCity | USA | OutsideUSA | In State |

LoginDuration | FewMinutes | FewDays | FewHours | ||

SoftwareUsed | PythonShell | Tableau | Hadoop | top |

We classify each user as one of three classes: y^{i}=Hacker, y^{i}=Student, or y^{i}=Faculty. Based on a

large training set, we wish to estimate all joint probability ** likelihoods**, e.g.,

P(x_{1}=Monthly,x_{2}=InCity,x_{3}=FewDays,x_{4}=top | y=Hacker), P(x_{1}=Daily,x_{2}=USA,x_{3}=FewHours,x_{4}=Tableau | y=Hacker).

- Assuming the features
**are not**independent, how many total parameters need to be estimated, accounting for classifying students, faculty, and hackers?

- Assuming the features
**are**independent, how many total parameters need to be estimated, accounting for classifying students, faculty, and hackers?

Now assume we only classify subjects using the first two features, and we replace the discrete feature values with numbers:

VisitFrequency | 0 (Never) | 1 (Monthly) | 2 (Daily) | ||

LoginLocation | 0 (OnCampus) | 1 (InCity ) | 2 (USA) | 3 (OutsideUSA) | 4 (In State) |

We use a joint Gaussian likelihood for the probability of the two features for each class y:

P(x_{1},x_{2}|y) , and we also will estimate a prior probability for each of the three user classes.

- Assuming x
_{1}and x_{2}are__not independent__, how many parameters need to be learned to compute the likelihood P(x_{1}, x_{2}| y) - Assuming x
_{1}and x_{2}are__independent__, how many parameters need to be learned to compute the posterior probability P(y | x_{1}, x_{2})

- I have written a classifier to determine if my dog is sick or healthy. I record the sounds my dog makes once a minute for six minutes, obtaining six sound measurements s
_{1}, s_{2}, … , s_{6}. At each time, there is a likelihood my dog will make the sounds: bark, no-sound, pant, whimper:

S | P(s|y=sick) | P(s|y=healthy) |

Bark
No-sound Pant |
0.15 | 0.4 |

0.4 | 0.3 | |

0.25 | 0.1 |

We compute P(y | s_{1}, …, s_{6}) = P(y) ∏_{𝑗 }𝑃(𝑠_{𝑗}|𝑦)

Note, P(y=healthy)=0.8

Provide the y=sick and y=healthy likelihoods and the max-posterior classification for each of the following sound recordings

- s
_{1}= No-sound , s_{2}= Bark , s_{3}= No-sound, s_{4}= Bark, s_{5}= Bark, s_{6}= Pant

- s
_{1}= Bark, s_{2}= Bark , s_{3}= Whimper, s_{4}= No-sound, s_{5}= Bark, s_{6}= No-sound

- s
_{1}= Whimper, s_{2}= Whimper, s_{3}= Bark, s_{4}= No-sound, s_{5}= Bark, s_{6}= No-sound

- In lecture, we used the Gaussian probability function to express the likelihood of lightintensity conditioned on the weather being Cloudy, Eclipse, or Non-cloudy (clear skies). However, the Gaussian function allows for both positive
**and negative**light intensities. Let us instead consider the Rayleigh probability as the likelihood for P(light|weather):

𝑙𝑖𝑔ℎ𝑡 −𝑙𝑖𝑔ℎ𝑡2/(2𝜎𝑤2) 𝑃(𝑙𝑖𝑔ℎ𝑡|𝑤) = 𝜎_{𝑤}2 𝑒

This function **only** is defined for light ≥ 0 . It has one parameter, 𝜎_{𝑤}, determined by weather.

- Assuming 𝜎
_{𝑤 }= 3**2**, compute:- 𝑃(𝑙𝑖𝑔ℎ𝑡 = 1|𝜎
_{𝑤 }= 2) - 𝑃(𝑙𝑖𝑔ℎ𝑡 = 3|𝜎
_{𝑤 }= 2)

- 𝑃(𝑙𝑖𝑔ℎ𝑡 = 1|𝜎

- We have 200 measurements of light-intensities 𝑙𝑖𝑔ℎ𝑡
^{1}, 𝑙𝑖𝑔ℎ𝑡^{2}, ⋯ , 𝑙𝑖𝑔ℎ𝑡^{200}during a snow storm and we wish to estimate the corresponding parameter 𝜎_{𝑠𝑛𝑜𝑤}.**Derive the maximum likelihood estimate estimate for**𝝈_{𝒔𝒏𝒐𝒘}. Start with 𝑃(𝐷|𝜃) = ∏ 𝑃(𝑙𝑖𝑔ℎ𝑡^{𝑖}|𝜎_{𝑠𝑛𝑜𝑤})

𝑖

**Show at least three mathematical steps to get your estimate of **𝝈_{𝒔𝒏𝒐𝒘}**. **

- Let us assume we have a prior probability on 𝜎 values defined by the “half-normal distribution” P(𝜎):

𝜎2

𝑃 𝜎≥0

This function **only** is defined for 𝜎 ≥ 0.

Technical note: For simplicity I use a very specific version of the half-normal distribution here with no extra hyper-parameters. In this case, we are computing the value of 𝜎, not using it to control the shape of the half-normal distribution.

**Derive the maximum a posteriori estimate for **𝝈_{𝒔𝒏𝒐𝒘}**. Show at least three mathematical steps to get your estimate of **𝝈_{𝒔𝒏𝒐𝒘}

- You have developed a program that determines from a user’s movie-watching history whether s/he is an adult or a teenager. We know 25% of users are teenagers. If user X is a teenager, the program will say so with probability 90%. If user X is an adult, the program will say s/he is an adult with probability 60%.

Assume that the program says user X is a teenager. What is the probability s/he is actually an adult?

** **

𝑥^{𝑇}𝐴𝑥.

- In class we discussed a family of distributions with input between 0 and 1; this family was from the Beta distribution (𝑥|𝛼, 𝛽) :

𝑥𝛼−1(1 − 𝑥)𝛽−1

𝑃(𝑥|𝛼, 𝛽) =

𝐵(𝛼, 𝛽)

Where 0 ≤ 𝑥 ≤ 1

- What 𝛼, 𝛽 combination will produce each of the probabilities below. (At least three of the options below are valid) Option I : 𝛼 = 10, 𝛽 = 3

Option II : 𝛼 = 6, 𝛽 = 3

Option III : 𝛼 = 5, 𝛽 = 5

Option IV : 𝛼 = 30, 𝛽 = 9

Option V: 𝛼 = 1, 𝛽 = 500

- The Beta distribution 𝑃(𝑥|𝛼 = 20, 𝛽 = 5) reaches a maximum probability
*P*when x=x_{max}_{max}. How can we define a distribution that also reaches maximum probability when x=x_{max}, but the maximum probability is now larger (e.g., max probability is now 3P_{max}, where*P*was defined in the past sentence)._{max}

** **

**Programming**

__Detailed submission instructions:__ Code must be left in your private/CIS5800/ directory. Include all function definitions and your answers to questions 1 and 6 (as comments) in the file hw1.py . For this homework, we will require several numpy array inputs.

__Now, on to the programming__:

In class, we discussed classification using Max Likelihood and using Max Posterior. For this assignment, you will create the Max Posterior/Bayes classifier to label online social network users based on their posting history. Specifically, you will determine if each user is based in Cairo, Frankfurt, Philadelphia, or Seoul. You will make this determination based on a single feature x – the time the user most frequently posts online.

**Accessing our data **

The file hw1data.mat is available on our website (and on erdos using

cp ~dleeds/MLpublic/hw1data.mat .) Load this file into your Python session to get access to the trainData and testData numpy arrays . For each array, each row is one example data point. The first column represents the user class – 0 for Cairo, 1 for Frankfurt, 2 Philadelphia, and 3 Seoul – and the second column represents the corresponding postingTime (most common posting time) for the example data point (user).

Note postingTime will be determined based on the current time in New York City. Also, time will be recorded on the 24-hour clock, where 0 is midnight, 430 is 4:30am, and 1750 is 5:50pm.

**Programming assignments: **

- Inspect the distribution of the postingTime feature for each class and determine if it follows a Gaussian or a Uniform distribution. (Note, uniform was shown earlier in Lecture 1.) Record this result as a comment in hw1.py .

**You can inspect the distribution of values in a list/vector of numbers ****vector through a histogram. **

**import matplotlib.pyplot as plt plt.hist(vector) plt.show() **

*Regardless of our results from question 1, we will assume all distributions really are Gaussian for the rest of this assignment. *

- Write a function called learnParams that takes in a data set and returns the learned mean and standard deviation for each class. Specifically, the function will be called as:

params=learnMean(Data)

where Data is a numpy array with shape (N,2) where N is the number of data points and params is a numpy array with shape (M,2) where there are M classes, params[i,0] is the mean for class i and params[i,1] is the standard deviation of class i. learnParams(np.array([[0,200],[1,1500],[0,300],[1,1700],

[0,400],[1,1300]])

would return np.array([[300,100],[1500,200]])

- Write a function called learnPriors that takes in a data set and returns the prior probability of each class. Specifically, the function will be called as: priors=learnPriors(Data)

where Data is a numpy array with shape (N,2) where N is the number of data points and priors is a numpy array with shape (M) where there are M classes, priors[i] is the estimated prior probability for class i .

learnPriors(np.array([[0,200],[1,1500],[0,300],[1,1700],

[0,400],[1,1300]])

would return np.array([0.5,0.5])

- Write a function called labelBayes that takes in posting times for multiple users as well as the learned parameters for the likelihoods and prior, and return the most probably class for each user. Specifically, the function will be called as:

labelsOut = labelBayes(postTimes,paramsL,priors)

where postTimes is a numpy array of shape (K) containing post times for K users, paramsL is a numpy array with shape (M,2) matching the description of the output for learnParams and priors is a numpy array with shape (M) matching the description of the output for learnPriors ; labelsOut is a numpy array with shape (K) containing the most probable label for each user, where labelsOut[j] corresponds to postTimes[j] . Labels are computed using the Gaussian Bayes classifier!

** **

labelBayes(np.array([430,2110,845]),

np.array([[300,100],[1500,250]]),np.array([0.2,0.8]))

would return np.array([0,1,1])

** **

** **

- Write a function called evaluateBayes that takes in classifier parameters for likelihoods and priors, and a set of labels and feature values, and returns the percent of input data correctly classified. Specifically, the function will be called as:

accuracy = evaluateBayes(paramsL,priors,testData)

where paramsL is a numpy array with shape (M,2) matching the description of the output for learnParams and priors is a numpy array with shape (M) matching the description of the output for learnPriors , testData is a numpy array with shape (J,2) where testData[j,0] contains the label of data point j and testData[j,1] contains the feature value (posting time) for data point j ; accuracy is a number between 0 and 1 indicating the accuracy of the Gaussian Bayes classifier using the specified parameters on the specified input data set.

evaluateBayes(np.array([[300,100],[1500,200]]), np.array([0.2,0.8]),

np.array([0,430],[1,2110],[0,845]))

would return 0.6666

- Our definition for time-of-day for user posting is not truly linear or continuous. 859 (8:59am) is followed by 900 (9:00am), skipping the integers 860, 861, through 899. 2359 (11:59pm) is followed by 0 (midnight) – it is much closer in time to midnight than it is to 2030 (8:30pm), while the integer 2359 is much closer to 2030 than it is to 0.

Rewrite either learnParams (from question 2) or labelBayes (from question 4) to morenaturally reflect the circular nature of the clock, and to account for skips in integers for each hour. Call this function learnParamsClock or labelBayesClock.

**Explain the reasoning of your approach in a comment in your function. **

**There are many reasonable ways to answer this question! **

** **