Unit 3 - Practice Quiz

Supervised learning is characterized by its use of labeled datasets. The algorithm learns from input-output pairs to make predictions on new, unseen data.

The mode is the value that appears most often in a set of data. The mean is the average, and the median is the middle value when the data is sorted.

A six-sided die has three even numbers (2, 4, 6) out of six possible outcomes. Therefore, the probability is 3/6, which simplifies to 1/2.

A vector is a 1-dimensional array of numbers, which is a convenient way to represent the features of a single observation or data point.

Cross-validation is a technique for evaluating a model's performance by training it on subsets of the data and testing it on the complementary subset, giving a more robust estimate of its performance on new data.

This task is known as clustering, where the goal is to find hidden patterns or structures in unlabeled data. Since there are no predefined labels for the customer groups, it is an unsupervised learning problem.

The posterior probability, often written as , is the revised probability of a hypothesis H after observing evidence E. It's the result of applying Bayes' theorem.

A matrix is a rectangular array of numbers arranged in rows and columns. It's the standard way to represent a tabular dataset in machine learning.

Precision asks the question: 'Of all the instances the model predicted as positive, how many were actually positive?'. It is calculated as , where TP is True Positives and FP is False Positives.

Reinforcement learning involves an 'agent' learning to make decisions by performing actions in an environment to maximize a cumulative reward. This trial-and-error approach with rewards and penalties is the core of RL.

To find the median, you must first sort the numbers: [1, 3, 5, 7, 9]. The median is the middle value in the sorted list, which is 5.

A Bayesian Network is a probabilistic graphical model. Each node (or vertex) in the graph represents a random variable, and the edges represent conditional dependencies between them.

Probability is a measure of the likelihood of an event occurring. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain. It cannot be negative or greater than 1.

Feature engineering is a crucial pre-modeling step that uses domain knowledge to create features that make machine learning algorithms work better. It directly impacts model performance.

Regression is a type of supervised learning task where the goal is to predict a continuous numerical value (like a price), rather than a discrete category.

A scalar is simply an ordinary number (an integer or a real number). It is used to scale vectors and matrices through multiplication.

Classification is a supervised learning task where the model learns to assign a category or class label (e.g., 'spam' or 'not spam') to a new observation.

The range is the simplest measure of statistical dispersion or variability. It is calculated by subtracting the minimum value from the maximum value in a dataset.

The directed edges in a Bayesian Network represent conditional dependencies. An arrow from A to B means that the probability of B is dependent on the value of A. A is considered a 'parent' of B.

A False Positive (also called a Type I error) is an outcome where the model incorrectly predicts the positive class. In this case, 'having the disease' is the positive class, so the test falsely identifies a healthy person as being sick.

High precision (True Positives / (True Positives + False Positives)) means that when the model predicts a positive case, it is very likely to be correct. However, low recall (True Positives / (True Positives + False Negatives)) means the model fails to identify a large number of actual positive cases (it has many false negatives). This is a critical issue in medical diagnosis where failing to detect a disease is often more dangerous than a false alarm.

This is a classic clustering problem. Since there are no predefined labels for the customer segments, the goal is to discover inherent structures or patterns in the data. Unsupervised learning algorithms, such as K-Means or DBSCAN, are designed for this purpose.

PCA works by finding a new set of orthogonal axes, called principal components, that align with the directions of maximum variance in the dataset. These directions are defined by the eigenvectors of the covariance matrix. The first principal component corresponds to the eigenvector with the largest eigenvalue, capturing the most variance.

This is a classic example of the base rate fallacy, solved using Bayes' theorem. Because the disease is so rare, the number of false positives from the large healthy population (9,999 out of 10,000 people will generate about 100 false positives) will far outnumber the true positives from the small sick population (1 out of 10,000 will likely generate 1 true positive). Therefore, a positive test result is more likely to be a false positive than a true positive.

The mean is highly sensitive to outliers and extreme values. In a right-skewed distribution, the mean is pulled upwards by the high values, making it an inflated representation of the central tendency. The median, which is the middle value when data is sorted, is robust to outliers and will give a much better indication of the typical salary.

This problem has all the key elements of reinforcement learning: an agent (the program) interacting with an environment (the chess game), taking actions (making moves), and receiving delayed rewards (the outcome of the game). The agent learns a policy (a strategy for choosing moves) to maximize its long-term reward.

The training set is used to fit the model. The validation set is used to evaluate different hyperparameter settings and choose the best-performing model. The test set is kept completely separate until the very end to provide a final, unbiased estimate of how the chosen model will perform on new, unseen data. Using the test set for hyperparameter tuning would lead to an overly optimistic performance estimate.

Cosine similarity measures the cosine of the angle between two vectors. In the context of word embeddings, vectors for semantically similar words (e.g., 'king' and 'queen') are designed to point in similar directions. A cosine similarity close to 1 indicates a small angle and thus high semantic similarity, while a value close to 0 indicates orthogonality or dissimilarity.

Logistic regression models the probability that an input belongs to a particular class. The sigmoid function at its output squashes any real-valued number into the range [0, 1], which is interpreted as a probability. A value of 0.7 means the model estimates a 70% probability that the sample belongs to the positive class (class 1).

The arrow from Disease to Symptom indicates a direct causal or influential relationship, where the state of Disease affects the probability of Symptom. The network encodes the conditional probability . Observing that the patient has the disease means we use this conditional probability table to update our belief about the likelihood of the symptom.

KNN calculates distances (like Euclidean distance) between data points. If one feature has a much larger scale than others (e.g., salary in dollars vs. years of experience), its contribution to the distance calculation will overwhelm the others. Standardization rescales features to have a mean of 0 and a standard deviation of 1, ensuring that all features contribute more equally to the distance metric.

Let's apply the transformation to a basis vector, e.g., . The result is . The vector on the x-axis is rotated to the y-axis. Applying it to gives . This consistent behavior demonstrates a 90-degree counter-clockwise rotation around the origin.

In k-fold cross-validation, the dataset is divided into k equal (or nearly equal) folds. For each iteration, one fold is used as the validation set, and the remaining k-1 folds are used as the training set. With 1000 instances and k=5, each fold has 1000/5 = 200 instances. Therefore, the training set in each iteration will consist of 4 folds, which is 4 * 200 = 800 instances.

Classification tasks involve predicting a discrete, categorical label (e.g., yes/no, spam/not spam, flower species). Regression tasks involve predicting a continuous numerical value. Predicting a house price, which can be any value within a range, is a quintessential regression problem.

The general formula for the union of two events is . For mutually exclusive events, they cannot occur at the same time, so their joint probability is 0. Therefore, the formula simplifies to .

The goal of a spam filter is to determine the probability that an email is spam given that we have observed some evidence (like the word 'free'). This is a conditional probability, specifically the posterior probability . Bayes' theorem provides the framework to calculate this using the prior probability P(Spam), the likelihood P(contains 'free' | Spam), and the evidence P(contains 'free').

The test set must be held out and used only once for a final, unbiased evaluation. If you use the test set to guide your feature selection (or any other model tuning), you are implicitly fitting your model to the test set. The resulting performance metric will be inflated because the model has already 'seen' the data in some form, and it will not generalize as well to truly new, unseen data.

The determinant of a matrix represents the factor by which area (in 2D) or volume (in 3D) is scaled by the transformation. A determinant of zero means that the area/volume becomes zero. This happens when the transformation squashes the data onto a line (from 2D) or a plane (from 3D), effectively reducing its dimensionality and making the transformation non-invertible.

One of the core assumptions of linear regression is homoscedasticity, which means the variance of the residuals (errors) should be constant across all levels of the independent variables. A funnel shape in the residual plot indicates that the error variance is not constant (it increases or decreases with the fitted values), which is known as heteroscedasticity. This can affect the reliability of the model's coefficient estimates and significance tests.

Semi-supervised learning is a paradigm that falls between supervised and unsupervised learning. It is used for problems where you have a small amount of labeled data and a large amount of unlabeled data. The goal is to leverage the structure within the unlabeled data to improve the performance of a model that is initially trained on the small labeled set.

A diagonal covariance matrix indicates that the original features are already uncorrelated. The goal of PCA is to find a new, orthogonal basis of uncorrelated variables (the principal components). Since the original feature axes already form an orthogonal basis of uncorrelated variables, the principal components will be aligned with these axes. The transformation matrix will be a permutation matrix or the identity matrix, meaning there is no rotation of the data, only scaling based on the variances (the diagonal entries).

In a highly imbalanced dataset, a naive model predicting the majority class (non-fraudulent) would achieve 99.9% accuracy, making this metric useless. The AUC-PR is a more informative metric here. The baseline for AUC-PR (the score of a random classifier) is equal to the fraction of positives, which is 0.001. A score of 0.2 is 200 times better than random. Therefore, it correctly identifies that accuracy is misleading and that the model has learned a meaningful signal, even if its performance is not perfect.

This structure is known as a 'v-structure' or a 'collider'. Node B is a collider because two arrows point into it. In a v-structure, the path between A and C is blocked by default, making them marginally independent (). However, if we observe or condition on the collider B (or any of its descendants), the path becomes unblocked, and information can flow between A and C. This makes them conditionally dependent. This phenomenon is often called 'explaining away'.

This is a classic Reinforcement Learning problem due to the agent-environment interaction and reward-based learning. Supervised learning is not feasible as there is no labeled dataset of 'correct' movements. Unsupervised learning can help with state representation but doesn't solve the core control problem. Within RL, the dynamics of a bipedal robot on varied terrain are extremely complex and difficult to model accurately, making model-based RL challenging. The problem involves a continuous action space (joint torques/angles) and requires learning a stochastic policy, which makes model-free, policy-based methods like Proximal Policy Optimization (PPO) or Asynchronous Advantage Actor-Critic (A3C) the state-of-the-art and most suitable approach.

The matrix , known as the Gram matrix, is singular if and only if the columns of X (the features) are linearly dependent. This condition is called perfect multicollinearity. From a linear algebra perspective, if is singular, it does not have an inverse, meaning the normal equation does not yield a unique solution. Instead, it defines a system of linear equations with either no solution or infinite solutions. In the context of least squares, this corresponds to an infinite number of coefficient vectors that all achieve the same minimum squared error.

As K increases, the training set size for each fold () approaches the full dataset size N. Models trained on more data are generally better, so the performance estimate becomes less biased (it's a better estimate of the true performance of a model trained on N samples). However, with LOOCV, the N training sets are nearly identical (differing by only one sample). This high correlation between the models trained in each fold leads to a high variance in the final performance estimate. The average of highly correlated variables has a much higher variance than the average of independent variables. Thus, the trade-off is accepting higher variance for lower bias.

This requires a careful application of Bayes' theorem. Let D be having the disease, and T be testing positive. We want .
We are given:

(True Positive Rate)
(True Negative Rate), so (False Positive Rate)

Using Bayes' theorem:


This is approximately 0.49%. Even with a seemingly accurate test, the vast number of false positives from the healthy population swamps the true positives from the sick population due to the disease's rarity.

The core strength of off-policy learning is the decoupling of the target policy (the policy we want to learn) from the behavior policy (the policy used to generate experience). Q-Learning's update rule uses the max Q-value for the next state, effectively learning about the greedy (optimal) policy, regardless of which action was actually taken by the exploratory behavior policy. SARSA, an on-policy algorithm, updates its Q-values based on the action actually taken, thus learning the value of its current behavior policy (including its exploration steps). This decoupling allows off-policy methods to learn from historical data or from a human expert's actions, which is not possible with on-policy methods.

This is a non-intuitive result related to the base rates and evidence accumulation (Simpson's Paradox can be related). The probability depends on the likelihoods of seeing and for both classes ( and ). While both features individually increase the probability of , it is possible that the combination is extremely common when . If and are both very high (e.g., 0.9), their product under the independence assumption makes the likelihood very high. This strong evidence for can potentially outweigh the evidence for , pushing the posterior probability down, possibly even below 0.6.

The standard t-test assumes that the measurements (the performance differences per fold) are independent. However, in k-fold cross-validation, the training sets for any two folds overlap by a large amount. This means the models produced are not independent, and their performance scores are correlated. This correlation leads to an underestimation of the variance of the average difference. A smaller estimated variance makes the t-statistic larger, increasing the likelihood of rejecting the null hypothesis when it is true (a Type I error). Dietterich's seminal paper showed that this issue is particularly pronounced for standard k-fold CV and proposed the 5x2 CV t-test as a partial remedy, though the core issue of dependence remains a concern in ML model comparison.

K-Means is a hard-assignment algorithm that assumes clusters are isotropic (spherical) and of similar size. It can only find linear decision boundaries between clusters. GMM is a soft-assignment algorithm that generalizes K-Means. It can model non-spherical (elliptical) clusters by estimating a full covariance matrix for each component. The fact that GMM found overlapping, elliptical clusters suggests this is a more accurate representation of the underlying data structure than the simplistic, spherical assumption of K-Means. GMM's probabilistic assignments also provide a measure of uncertainty for points in the overlapping regions.

One-hot encoding a high-cardinality categorical feature like 'zip_code' leads to the curse of dimensionality, creating thousands of new binary features. This sparsity makes it difficult for a linear model to learn robust weights, especially for zip codes with few samples. A powerful alternative is target encoding (or mean encoding). This replaces the categorical feature with a single numerical feature representing the average target value for that category. This directly encodes information about the target variable, making it highly predictive. The risk is data leakage and overfitting if not implemented carefully (e.g., by calculating the means on the training set only and applying them to the validation/test sets, or by using a cross-validation scheme).

The Naive Bayes classifier calculates the posterior as . If and provide the same information (perfect correlation), including both and in the product is like including the same piece of evidence twice. This squaring effect on the likelihood term will artificially amplify the evidence, pushing the calculated posterior probabilities towards 0 or 1 and making the model overly confident in its predictions.

The reconstructed matrix provides the predictions. An individual entry (rating of user i for item j) is the dot product of the i-th row of and the j-th row of (which is the j-th column of ). Geometrically, this means we represent both users and items as vectors in a common k-dimensional latent space. A high rating is predicted when the user and item vectors are closely aligned and have large magnitudes in this space, as captured by the dot product.

Overfitting in an RBF SVM indicates the decision boundary is too complex and sensitive to individual data points.

1. The C parameter is the regularization parameter. A high C value penalizes misclassifications heavily, leading to a complex, tight-fitting boundary. Decreasing C allows for a 'softer' margin, tolerating more misclassifications in the training set to achieve a simpler, more generalizable decision boundary.
2. The gamma parameter defines the influence of a single training example. A high gamma leads to a very localized influence, creating a complex, 'spiky' boundary. Decreasing gamma makes the influence of each support vector broader, resulting in a smoother, less complex boundary. Both actions serve to regularize the model and combat overfitting.

The ROC curve plots True Positive Rate (TPR) vs. False Positive Rate (FPR). FPR is calculated as . In a highly imbalanced dataset, the number of True Negatives (TN) is massive. A model can generate a large number of False Positives (FP) without making the FPR significantly large, resulting in a deceptively optimistic AUC-ROC score. The Precision-Recall curve, however, plots Precision () vs. Recall (TPR). It does not use TN in its calculation. It focuses directly on the model's performance on the minority (positive) class, making it a much more informative metric for tasks like fraud or disease detection.

This structure is a 'chain'. According to the rules of d-separation, a path is blocked if it contains a chain () where the middle node (B) is conditioned on. Intuitively, all information from X that influences Z must pass through Y. Once the state of Y is known, X provides no additional information about Z. Therefore, conditioning on Y renders X and Z conditionally independent.

The solution for Ridge Regression is . Multicollinearity causes the matrix to be nearly singular (ill-conditioned), meaning some of its eigenvalues are close to zero. The inversion of such a matrix is numerically unstable. By adding (a positive constant times the identity matrix), we are effectively adding to each eigenvalue of . This shifts all eigenvalues away from zero, guaranteeing that the matrix is invertible and well-conditioned, thus stabilizing the solution. This process shrinks the resulting coefficients, reducing their variance.

The key distinction is the source and purpose of the 'extra' data/knowledge. Semi-supervised learning leverages unlabeled data from the target domain to better understand its underlying structure, helping the model generalize from the few labeled examples it has for that same domain. Transfer learning leverages a model (and its learned features) trained on a different, often much larger, dataset (e.g., ImageNet) and applies it to a new target task (e.g., medical image classification), which may have limited labeled data. It's about transferring knowledge across tasks, not leveraging unlabeled data for the same task.

The most critical flaw is data leakage. Time-series data has an inherent temporal order. By randomly shuffling, a fold's training set could contain data from 2022, while its validation set contains data from 2021. The model learns from information that would not have been available at the time of the 'prediction', a situation that is impossible in a real-world deployment. This leakage leads to performance metrics that are artificially inflated and do not reflect the model's true ability to forecast the future. The correct approach is to use a method that respects temporal order, such as walk-forward validation or time-series split.