.. _sec_naive_bayes:
Bayes ngây thơ
==============
Trong suốt các phần trước, chúng ta đã học về lý thuyết xác suất và các
biến ngẫu nhiên. Để đưa lý thuyết này vào hoạt động, chúng ta hãy giới
thiệu bộ phân loại Bayes\* ngây thơ. Điều này không sử dụng gì ngoài các
nguyên tắc cơ bản xác suất để cho phép chúng tôi thực hiện phân loại các
chữ số.
Học tập là tất cả về việc đưa ra các giả định. Nếu chúng ta muốn phân
loại một ví dụ dữ liệu mới mà chúng ta chưa bao giờ thấy trước đây chúng
ta phải đưa ra một số giả định về ví dụ dữ liệu nào tương tự nhau. Phân
loại Bayes ngây thơ, một thuật toán phổ biến và rõ ràng đáng kể, giả
định tất cả các tính năng đều độc lập với nhau để đơn giản hóa việc tính
toán. Trong phần này, chúng tôi sẽ áp dụng mô hình này để nhận dạng các
ký tự trong hình ảnh.
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%matplotlib inline
import math
from mxnet import gluon, np, npx
from d2l import mxnet as d2l
npx.set_np()
d2l.use_svg_display()
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%matplotlib inline
import math
import torch
import torchvision
from d2l import torch as d2l
d2l.use_svg_display()
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%matplotlib inline
import math
import tensorflow as tf
from d2l import tensorflow as d2l
d2l.use_svg_display()
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def transform(data, label):
return np.floor(data.astype('float32') / 128).squeeze(axis=-1), label
mnist_train = gluon.data.vision.MNIST(train=True, transform=transform)
mnist_test = gluon.data.vision.MNIST(train=False, transform=transform)
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data_transform = torchvision.transforms.Compose([
torchvision.transforms.ToTensor(),
lambda x: torch.floor(x * 255 / 128).squeeze(dim=0)
])
mnist_train = torchvision.datasets.MNIST(
root='./temp', train=True, transform=data_transform, download=True)
mnist_test = torchvision.datasets.MNIST(
root='./temp', train=False, transform=data_transform, download=True)
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0.6%Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz to ./temp/MNIST/raw/train-images-idx3-ubyte.gz
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Extracting ./temp/MNIST/raw/train-images-idx3-ubyte.gz to ./temp/MNIST/raw
102.8%
Downloading http://yann.lecun.com/exdb/mnist/train-labels-idx1-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/train-labels-idx1-ubyte.gz to ./temp/MNIST/raw/train-labels-idx1-ubyte.gz
Extracting ./temp/MNIST/raw/train-labels-idx1-ubyte.gz to ./temp/MNIST/raw
Downloading http://yann.lecun.com/exdb/mnist/t10k-images-idx3-ubyte.gz
8.0%Downloading http://yann.lecun.com/exdb/mnist/t10k-images-idx3-ubyte.gz to ./temp/MNIST/raw/t10k-images-idx3-ubyte.gz
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Downloading http://yann.lecun.com/exdb/mnist/t10k-labels-idx1-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/t10k-labels-idx1-ubyte.gz to ./temp/MNIST/raw/t10k-labels-idx1-ubyte.gz
112.7%Extracting ./temp/MNIST/raw/t10k-labels-idx1-ubyte.gz to ./temp/MNIST/raw
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((train_images, train_labels), (
test_images, test_labels)) = tf.keras.datasets.mnist.load_data()
# Original pixel values of MNIST range from 0-255 (as the digits are stored as
# uint8). For this section, pixel values that are greater than 128 (in the
# original image) are converted to 1 and values that are less than 128 are
# converted to 0. See section 18.9.2 and 18.9.3 for why
train_images = tf.floor(tf.constant(train_images / 128, dtype = tf.float32))
test_images = tf.floor(tf.constant(test_images / 128, dtype = tf.float32))
train_labels = tf.constant(train_labels, dtype = tf.int32)
test_labels = tf.constant(test_labels, dtype = tf.int32)
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image, label = mnist_train[2]
image.shape, label
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((28, 28), array(4, dtype=int32))
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image, label = mnist_train[2]
image.shape, label
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(torch.Size([28, 28]), 4)
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image, label = train_images[2], train_labels[2]
image.shape, label.numpy()
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(TensorShape([28, 28]), 4)
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image.shape, image.dtype
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((28, 28), dtype('float32'))
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image.shape, image.dtype
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(torch.Size([28, 28]), torch.float32)
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image.shape, image.dtype
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(TensorShape([28, 28]), tf.float32)
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label, type(label), label.dtype
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(array(4, dtype=int32), mxnet.numpy.ndarray, dtype('int32'))
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label, type(label)
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(4, int)
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label.numpy(), label.dtype
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(4, tf.int32)
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images, labels = mnist_train[10:38]
images.shape, labels.shape
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((28, 28, 28), (28,))
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images = torch.stack([mnist_train[i][0] for i in range(10, 38)], dim=0)
labels = torch.tensor([mnist_train[i][1] for i in range(10, 38)])
images.shape, labels.shape
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(torch.Size([28, 28, 28]), torch.Size([28]))
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images = tf.stack([train_images[i] for i in range(10, 38)], axis=0)
labels = tf.constant([train_labels[i].numpy() for i in range(10, 38)])
images.shape, labels.shape
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(TensorShape([28, 28, 28]), TensorShape([28]))
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d2l.show_images(images, 2, 9);
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d2l.show_images(images, 2, 9);
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d2l.show_images(images, 2, 9);
.. figure:: output_naive-bayes_6e475d_81_0.svg
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X, Y = mnist_train[:] # All training examples
n_y = np.zeros((10))
for y in range(10):
n_y[y] = (Y == y).sum()
P_y = n_y / n_y.sum()
P_y
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array([0.09871667, 0.11236667, 0.0993 , 0.10218333, 0.09736667,
0.09035 , 0.09863333, 0.10441667, 0.09751666, 0.09915 ])
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X = torch.stack([mnist_train[i][0] for i in range(len(mnist_train))], dim=0)
Y = torch.tensor([mnist_train[i][1] for i in range(len(mnist_train))])
n_y = torch.zeros(10)
for y in range(10):
n_y[y] = (Y == y).sum()
P_y = n_y / n_y.sum()
P_y
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tensor([0.0987, 0.1124, 0.0993, 0.1022, 0.0974, 0.0904, 0.0986, 0.1044, 0.0975,
0.0992])
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X = train_images
Y = train_labels
n_y = tf.Variable(tf.zeros(10))
for y in range(10):
n_y[y].assign(tf.reduce_sum(tf.cast(Y == y, tf.float32)))
P_y = n_y / tf.reduce_sum(n_y)
P_y
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n_x = np.zeros((10, 28, 28))
for y in range(10):
n_x[y] = np.array(X.asnumpy()[Y.asnumpy() == y].sum(axis=0))
P_xy = (n_x + 1) / (n_y + 1).reshape(10, 1, 1)
d2l.show_images(P_xy, 2, 5);
.. figure:: output_naive-bayes_6e475d_99_0.svg
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n_x = torch.zeros((10, 28, 28))
for y in range(10):
n_x[y] = torch.tensor(X.numpy()[Y.numpy() == y].sum(axis=0))
P_xy = (n_x + 1) / (n_y + 1).reshape(10, 1, 1)
d2l.show_images(P_xy, 2, 5);
.. figure:: output_naive-bayes_6e475d_102_0.svg
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n_x = tf.Variable(tf.zeros((10, 28, 28)))
for y in range(10):
n_x[y].assign(tf.cast(tf.reduce_sum(
X.numpy()[Y.numpy() == y], axis=0), tf.float32))
P_xy = (n_x + 1) / tf.reshape((n_y + 1), (10, 1, 1))
d2l.show_images(P_xy, 2, 5);
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def bayes_pred(x):
x = np.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = p_xy.reshape(10, -1).prod(axis=1) # p(x|y)
return np.array(p_xy) * P_y
image, label = mnist_test[0]
bayes_pred(image)
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array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
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def bayes_pred(x):
x = x.unsqueeze(0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = p_xy.reshape(10, -1).prod(dim=1) # p(x|y)
return p_xy * P_y
image, label = mnist_test[0]
bayes_pred(image)
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tensor([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
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def bayes_pred(x):
x = tf.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = tf.math.reduce_prod(tf.reshape(p_xy, (10, -1)), axis=1) # p(x|y)
return p_xy * P_y
image, label = train_images[0], train_labels[0]
bayes_pred(image)
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*math.log(a))
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underflow: 0.0
logarithm is normal: -1805.2267129073316
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*math.log(a))
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underflow: 0.0
logarithm is normal: -1805.2267129073316
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*tf.math.log(a).numpy())
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underflow: 0.0
logarithm is normal: -1805.2267379760742
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log_P_xy = np.log(P_xy)
log_P_xy_neg = np.log(1 - P_xy)
log_P_y = np.log(P_y)
def bayes_pred_stable(x):
x = np.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = p_xy.reshape(10, -1).sum(axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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:class: output
array([-269.0042 , -301.73447, -245.21458, -218.8941 , -193.46907,
-206.10315, -292.54315, -114.62834, -220.35619, -163.18881])
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log_P_xy = torch.log(P_xy)
log_P_xy_neg = torch.log(1 - P_xy)
log_P_y = torch.log(P_y)
def bayes_pred_stable(x):
x = x.unsqueeze(0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = p_xy.reshape(10, -1).sum(axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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tensor([-269.0042, -301.7345, -245.2146, -218.8941, -193.4691, -206.1031,
-292.5432, -114.6283, -220.3562, -163.1888])
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log_P_xy = tf.math.log(P_xy)
log_P_xy_neg = tf.math.log(1 - P_xy)
log_P_y = tf.math.log(P_y)
def bayes_pred_stable(x):
x = tf.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = tf.math.reduce_sum(tf.reshape(p_xy, (10, -1)), axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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# Convert label which is a scalar tensor of int32 dtype to a Python scalar
# integer for comparison
py.argmax(axis=0) == int(label)
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array(True)
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py.argmax(dim=0) == label
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tensor(True)
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tf.argmax(py, axis=0, output_type = tf.int32) == label
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def predict(X):
return [bayes_pred_stable(x).argmax(axis=0).astype(np.int32) for x in X]
X, y = mnist_test[:18]
preds = predict(X)
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
.. figure:: output_naive-bayes_6e475d_159_0.svg
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def predict(X):
return [bayes_pred_stable(x).argmax(dim=0).type(torch.int32).item()
for x in X]
X = torch.stack([mnist_test[i][0] for i in range(18)], dim=0)
y = torch.tensor([mnist_test[i][1] for i in range(18)])
preds = predict(X)
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
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def predict(X):
return [tf.argmax(
bayes_pred_stable(x), axis=0, output_type = tf.int32).numpy()
for x in X]
X = tf.stack([train_images[i] for i in range(10, 38)], axis=0)
y = tf.constant([train_labels[i].numpy() for i in range(10, 38)])
preds = predict(X)
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
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X, y = mnist_test[:]
preds = np.array(predict(X), dtype=np.int32)
float((preds == y).sum()) / len(y) # Validation accuracy
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0.8426
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X = torch.stack([mnist_test[i][0] for i in range(len(mnist_test))], dim=0)
y = torch.tensor([mnist_test[i][1] for i in range(len(mnist_test))])
preds = torch.tensor(predict(X), dtype=torch.int32)
float((preds == y).sum()) / len(y) # Validation accuracy
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0.8426
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X = test_images
y = test_labels
preds = tf.constant(predict(X), dtype=tf.int32)
# Validation accuracy
tf.reduce_sum(tf.cast(preds == y, tf.float32)).numpy() / len(y)
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0.8426
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`Discussions `__
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`Discussions `__
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`Discussions `__
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