Machine-Learning- Homework 2 Solved

Description :  

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

continuousfeatures.

           Input:

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 chararrary to store ana image.

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

Training lable data from MNIST.
Testing image from MNIST
Testing label from MNISTToggle 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:

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.

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

 

40
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
 
41
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
 
42
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
 
43
 
44
 
45
 
46
 
47
 
48
 
49
 
50
 
51
 
52
 
53

54

55
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
 
56
 
57

58

59

60

61
 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 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
 
62
 
63
 
64
 
65
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
66
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
67
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
68
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
69
 
70

71

72

73
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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
 
74
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
 
75
 
76
 
77
 
78
 
79
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
 
80
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
 
81
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
 
82
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
 
83
 
84
 
85

86

87
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
88
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 92                                                                                                                                                                                                                                                                                                                       
         93 Error rate: 0.1535                                                                                          

 

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:

        

 

parameter a for the initial beta prior
parameter b for the initial beta prior
                                  Output: Print out the Binomial likelihood (based on MLE, of course), Beta prior and posterior

 

5

6
 

case 2: 0110101
 
7
Likelihood: 0.29375515303997485
 
8

9 10

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

case 3: 010110101101
 
12
Likelihood: 0.2286054241794335
 
13

14

15

16

17

18
   Beta prior:      a = 15 b = 14
 
Beta posterior: a = 22 b = 19
 
 
 
 
 
case 4: 0101101011101011010
 
Likelihood: 0.18286870706509092
 
   Beta prior:      a = 22 b = 19
 
19

20
Beta posterior: a = 33 b = 27
 
 
 
case 5: 111101100011110 Likelihood: 0.2143070548857833

Beta prior:       a = 33 b = 27 Beta posterior: a = 43 b = 32
 
21
 
22
 
23
 
24

25

26
 
 
 
 
 
case 6: 101110111000110
 
27
Likelihood: 0.20659760529408
 
28
   Beta prior:      a = 43 b = 32
 
29
Beta posterior: a = 52 b = 38

 

case 7: 1010010111
 
30
 
31
 
32
Likelihood: 0.25082265600000003
 
33
   Beta prior:      a = 52 b = 38
 
34
Beta posterior: a = 58 b = 42
 
35

36

37
 
 
case 8: 11101110110
 
Likelihood: 0.2619678932864457

   Beta prior:      a = 58 b = 42

Beta posterior: a = 66 b = 45

 

case 9: 01000111101
 
38
 
39
 
40
 
41
 
42

43

44

45

46
Likelihood: 0.23609128871506807
 
   Beta prior:      a = 66 b = 45
 
Beta posterior: a = 72 b = 50
 
 

case 10: 110100111
 
47
Likelihood: 0.27312909617436365
 
48

49

50

51

52
   Beta prior:      a = 72 b = 50
 
Beta posterior: a = 78 b = 53
 
 
 
 
 
case 11: 01101010111
 
Likelihood: 0.24384881449471862
 
53
   Beta prior:      a = 78 b = 53
 
3. 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 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).