Machine-Learning -  ML Homework2 - Solved

1.    Naive Bayes classifier       
Create a Naive Bayes classifier for each handwritten digit that support discrete and continuous features.

          Input:

1.    Training image data from MNIST

 You Must download the MNIST from this website and parse the data by yourself. (Please do not use the build in dataset or you'll not get 100.) Please read the description in the link to understand the format.

 Basically, each image is represented by             bits (Whole binary file is in big endian format; you need to deal with it), you can use a char arrary to store an image.

     There are some headers you need to deal with as well, please read the link for more details.

2.    Training lable data from MNIST.

3.    Testing image from MNIST

4.    Testing label from MNIST 5. Toggle option

 0: discrete mode

1: continuous mode

TRAINING SET IMAGE FILE (train-images-idx3-ubyte)                                                                 

offset
type
value
description
0000
32 bit integer
0x00000803(2051)
magic number
0004
32 bit integer
60000
number of images
0008
32 bit integer
28
number of rows
0012
32 bit integer
28
number of columns
0016
unsigned byte
??
pixel
0017
unsigned byte
??
pixel
...
...
...
...
xxxx
unsigned byte
??
pixel
TRAINING SET LABEL FILE (train-labels-idx1-ubyte)                                                                     

offset
type
value
description
0000
32 bit integer
0x00000801(2049)
magic number
0004
32 bit integer
60000
number of items
0008
unsigned byte
??
label
0009
unsigned byte
??
label
...
...
...
...
xxxx
unsigned byte
??
label
The labels values are from 0 to 9.

 Output:

Print out the the posterior (in log scale to avoid underflow) of the ten categories (0-9) for each image in INPUT 3. Don't forget to marginalize them so sum it up will equal to 1.

    For each test image, print out your prediction which is the category having the highest posterior, and tally the prediction by comparing with INPUT 4.

 Print out the imagination of numbers in your Bayes classifier

For each digit, print a   binary image which 0 represents a white pixel, and 1 represents a black pixel.

The pixel is 0 when Bayes classifier expect the pixel in this position should less then 128 in original image, otherwise is 1. Calculate and report the error rate in the end.

 Function:

1.    In Discrete mode:

     Tally the frequency of the values of each pixel into 32 bins. For example, The gray level 0 to 7 should be classified to bin 0, gray level 8 to 15 should be bin 1 ... etc. Then perform Naive Bayes classifier. Note that to avoid empty bin, you can use a peudocount (such as the minimum value in other bins) for instead.

2.    In Continuous mode:

     Use MLE to fit a Gaussian distribution for the value of each pixel. Perform Naive Bayes classifier.

 

 

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0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ... all other imagination of numbers goes here ...

9:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 

2.    Online learning        
Use online learning to learn the beta distribution of the parameter p (chance to see 1) of the coin tossing trails in batch.

          Input:

1.    A file contains many lines of binary outcomes:

 

2.    parameter a for the initial beta prior

3.    parameter b for the initial beta prior

    Output: Print out the Binomial likelihood (based on MLE, of course), Beta prior and posterior probability (parameters only) for each line.

 Function: Use Beta-Binomial conjugation to perform online learning.


 

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case 2: 0110101

Likelihood: 0.29375515303997485

Beta prior:     a = 11  b = 11 Beta posterior: a = 15  b = 14

case 3: 010110101101 Likelihood: 0.2286054241794335

Beta prior:     a = 15  b = 14 Beta posterior: a = 22  b = 19

case 4: 0101101011101011010 Likelihood: 0.18286870706509092

Beta prior:     a = 22  b = 19 Beta posterior: a = 33  b = 27

case 5: 111101100011110 Likelihood: 0.2143070548857833

Beta prior:     a = 33  b = 27 Beta posterior: a = 43  b = 32

case 6: 101110111000110 Likelihood: 0.20659760529408

Beta prior:     a = 43  b = 32 Beta posterior: a = 52  b = 38

case 7: 1010010111

Likelihood: 0.25082265600000003

Beta prior:     a = 52  b = 38 Beta posterior: a = 58  b = 42

case 8: 11101110110 Likelihood: 0.2619678932864457

Beta prior:     a = 58  b = 42 Beta posterior: a = 66  b = 45

case 9: 01000111101 Likelihood: 0.23609128871506807

Beta prior:     a = 66  b = 45 Beta posterior: a = 72  b = 50

case 10: 110100111

Likelihood: 0.27312909617436365

Beta prior:     a = 72  b = 50 Beta posterior: a = 78  b = 53

case 11: 01101010111

Likelihood: 0.24384881449471862

Beta prior:     a = 78  b = 53
 

 


3.   Show      the   distribution     of      online       learning    
Following the result of 2. Online learning, try to show distribution of prior, likelihood function and posterior step by step. 

  

For example, the prior is given by a beta distribution with parameters a=2, b=2, and the likelihood function, given with N=m=1, corresponds to a single observation of x=1, so that the posterior is given by a beta distribution with parameters a=3, b=2. 

 

4.   Prove     Beta-Binomial conjugation 
    Try to proof Beta-Binomial conjugation and write the process on paper. ※ You should write down the proof process on paper and take a picture. When you hand in HW02, it must contain your code and picture. 

 

             l     NOTE: 

¡  Use whatever programming language you prefer. 

¡  You can’t use numpy.random.beta in HW02. That would be great if you implement all distribution by yourself. 

¡  HW02 must contain your code and proof process (can be .pdf or any image format).