Ensemble Method DefinitionAn ensemble method is a machine learning approach that combines multiple weak learners (individual models) to produce a stronger and more accurate predictive model.The key idea is that by aggregating the predictions of several models, the ensemble can reduce variance, bias, or improve generalization compared to a single model. Representative Types of Ensemble MethodsVo..

In the previous post, we trained an MNIST network with the quadratic (MSE) cost and a per-sample backprop loop.In this post, we switch to the Cross-Entropy (CE) cost and a mini-batch (vectorized) update. With sigmoid + CE (binary) or softmax + CE (multiclass), the output error isThere is no extra σ′(z) factor multiplying the error at the output layer.This avoids the severe gradient shrinkage you..

This code is based on Michael Nielsen’s implementation and runs on the MNIST handwritten digit recognition dataset.The key components of this code are the class structure, feedforward process, stochastic gradient descent (SGD), and backpropagation.The code is as follows: def sigmoid(z): return 1.0 / (1.0 + np.exp(-z)) import numpy as np import random def sigmoid_prime(z): """Derivative of..

Before this chapter, we discussed Bayesian learning.The Bayesian formula is as follows:P(y=c∣x)=P(x∣y=c)⋅P(y=c)In this formula:P(x∣y=c) is called the likelihoodP(y=c) is called the priorLet’s dive into an example to better understand this concept.Suppose we want to predict the class label for a given input feature vector.For example, imagine the input features are:x=[0.5, 0.3, 0.7, 0.8]When this..

According to the definition of conditional probability,P(X = x | Y = y) = p(X = x, Y = y) / p(Y = y).We can rewrite the numerator using the product rule:p(X = x, Y = y) = p(X = x) × p(Y = y | X = x).We can also rewrite the denominator using the law of total probability:p(Y = y) = Σ over x' [ p(X = x') × p(Y = y | X = x') ].For example, if the event Y = y corresponds to "ate melon",then the total..

Let A be a linear transformation, where A ∈ R^(m×n).The set of all vectors x such that A x = 0 is called the null space of A.In geometric intuition, the null space represents the set of directions that get "squished" to the zero vector by A.For example, letA = [[1, 2], [3, 6]].In this case, the null space is{ [x, y]ᵗ ∈ R² | x = -2y },or equivalently,{ [-2y, y]ᵗ | y ∈ R }.One example from this sp..