Questions Mixture Models and Expectation-Maximization

(Page 1: Title Slide)

• Q1: What is the main topic of this lecture?

  • A1: The main topic is Mixture Models and the Expectation-Maximization (EM) algorithm.

(Page 2: Motivation)

• Q2: Why are latent (unobserved) random variables useful in modeling?

  • A2: Introducing latent variables can help express complex (marginal) distributions that might be difficult to model directly.

• Q3: What is a common example of a model that uses latent variables?

  • A3: Mixture models, particularly Gaussian Mixture Models (GMMs), are a common example.

• Q4: What are two main applications of mixture models?

  • A4:
    1. Clustering (unsupervised learning).
    2. Representing complex probability distributions.

• Q5: How can the parameters of mixture models be estimated?

  • A5: Parameters can be estimated using maximum-likelihood estimation techniques, such as the Expectation-Maximization (EM) algorithm.

(Pages 3-8: K-means Clustering)

• Q6: What is the goal of K-means clustering?

  • A6: The goal is to find $K$ cluster centers, denoted as $\{{\mu }_1,\dots ,{\mu }_K\}$, such that the sum of squared distances between each data point and its assigned cluster center is minimized.

• Q7: Define the objective function $J$ for K-means clustering.

  • A7: The objective function is: $J={\sum }_{n=1}^N{\sum }_{k=1}^K{r}_{nk}||x_n-{\mu }_k|{|}^2$ where $r_{nk}=1$ if $x_n$ is assigned to cluster $k$, and $0$ otherwise.

• Q8: Describe the iterative process of the K-means algorithm.

  • A8:
    1. Initialization: Start with initial values for the cluster centers $\{{\mu }_k\}$.
    2. Assignment Step: Assign each data point $x_n$ to the nearest cluster center, updating $r_{nk}$. $r_{nk}=\{\begin{array}{ll}1& \text{if }k={\text{arg min}}_j||x_n-{\mu }_j||\\ 0& \text{otherwise}\end{array}$
    3. Update Step: Update the cluster centers by computing the mean of the data points assigned to each cluster. ${\mu }_k=\frac{{\sum }_{n=1}^N{r}_{nk}{x}_n}{{\sum }_{n=1}^N{r}_{nk}}$
    4. Repeat: Iterate steps 2 and 3 until the cluster assignments (or cluster centers) no longer change.

• Q9: How are the cluster means updated in the K-means algorithm, given the assignments $r_{nk}$?

  • A9: ${\mu }_k$ is updated as the mean of all points assigned to cluster k: ${\mu }_k=\frac{{\sum }_{n=1}^N{r}_{nk}{x}_n}{{\sum }_{n=1}^N{r}_{nk}}$. This is derived by setting the derivative of J with respect to ${\mu }_k$ equal to 0.

(Page 9: 2D Example)

• Q10: What does the magenta line represent in K-means result?
  • A10: Decision Boundary.

(Page 10: The Cost Function)

• Q11: What happens to the cost function $J$ in K-means after each iteration?

  • A11: The cost function $J$ is minimized after every step (either assignment or update).

• Q12: What do the “blue steps” and “red steps” represent in the cost function plot?

  • A12: Blue steps represent updating the assignments ($r_{nk}$). Red steps represent updating the cluster means (${\mu }_k$).

(Page 11: K-Means for Segmentation)

• Q13: Give an example of a practical application of K-means clustering beyond simple data point grouping.

  • A13: Image segmentation, where pixels are grouped based on color similarity.

(Page 12: K-Means: Additional Remarks)

• Q14: Does K-means always converge to a global minimum?

  • A14: No, K-means always converges, but it is not guaranteed to find the global minimum. It can get stuck in local minima.

• Q15: What is the “online” version of K-means?

  • A15: In the online version, after each new data point $x_n$ is added, the nearest cluster center ${\mu }_k$ is updated immediately: ${\mu }_k^{\text{new}}={\mu }_k^{\text{old}}+{\eta }_n(x_n-{\mu }_k^{\text{old}})$ where ${\eta }_n$ is a learning rate.

• Q16: What is the K-medoid variant of K-means?

  • A16: K-medoid replaces the Euclidean distance with a general dissimilarity measure $V$. The objective function becomes: $\tilde{J}={\sum }_{n=1}^N{\sum }_{k=1}^K{r}_{nk}V(x_n,{\mu }_k)$

(Pages 13-14: Mixtures of Gaussians)

• Q17: What is a Gaussian Mixture Model (GMM)?

  • A17: A GMM assumes that the data is generated from a mixture of several Gaussian distributions. Each Gaussian represents a cluster.

• Q18: Define the probability density function $p(x)$ for a GMM.

  • A18: $p(x)={\sum }_{k=1}^{K}p(z_k=1)\mathcal{N}(x|{\mu }_k,{\Sigma }_k)={\sum }_{k=1}^K{\pi }_k\mathcal{N}(x|{\mu }_k,{\Sigma }_k)$ where:
    • $z_k$ is a binary latent variable indicating whether $x$ belongs to the $k$-th Gaussian component.
    • ${\pi }_k=p(z_k=1)$ is the mixing coefficient for the $k$-th Gaussian, representing the prior probability that a data point belongs to that component. ${\sum }_{k=1}^K{\pi }_k=1$ and ${\pi }_k\in [0,1]$.
    • $\mathcal{N}(x|{\mu }_k,{\Sigma }_k)$ is the Gaussian probability density function with mean ${\mu }_k$ and covariance matrix ${\Sigma }_k$.

• Q19: What are the constraints on the latent variable $z_k$ and the mixing coefficients ${\pi }_k$ in a GMM?

  • A19: $z_k\in \{0,1\}$ and ${\sum }_{k=1}^K{z}_k=1$ for a given data point. For the mixing coefficients, ${\sum }_{k=1}^K{\pi }_k=1$.

(Page 15: Parameter Estimation)

• Q20: What is the goal of parameter estimation in a GMM?

  • A20: The goal is to find the parameters (mixing coefficients ${\pi }_k$, means ${\mu }_k$, and covariance matrices ${\Sigma }_k$) that maximize the likelihood of the observed data.

• Q21: Write down the log-likelihood function for a GMM.

  • A21: The log-likelihood is: $\log p(X|\pi ,\mu ,\Sigma )={\sum }_{n=1}^N\log \left({\sum }_{k=1}^K{\pi }_k\mathcal{N}(x_n|{\mu }_k,{\Sigma }_k)\right)$

• Q22: Why is maximizing the log-likelihood for a GMM more difficult than for a single Gaussian?

  • A22: The presence of the summation inside the logarithm prevents a closed-form solution like the one found for a single Gaussian. The parameters are coupled in a complex way.

(Page 16-17: Problems with MLE for Gaussian Mixtures)

• Q23: Describe the overfitting problem that can occur when using maximum likelihood estimation (MLE) with GMMs.

  • A23: If a Gaussian component’s mean (${\mu }_k$) coincides with a single data point ($x_n$), and its covariance matrix is proportional to the identity matrix (${\Sigma }_k={\sigma }_k^{2}I$), the likelihood can become arbitrarily large as ${\sigma }_k$ approaches zero. This leads to a singularity and overfitting.

• Q24: What is the identifiability problem in GMMs?

  • A24: The order of the Gaussian components is arbitrary. For $K$ components, there are $K!$ equivalent solutions that yield the same likelihood, making the parameters not uniquely identifiable.

(Page 18: Expectation-Maximization)

• Q25: What is the Expectation-Maximization (EM) algorithm?

  • A25: EM is an iterative algorithm for finding maximum likelihood estimates in models with latent variables. It’s a general method, but it’s commonly used for GMMs.

• Q26: What is the main idea behind the EM algorithm?

  • A26: The main idea is to iteratively estimate both the model parameters and the values of the latent variables. It alternates between an Expectation (E) step and a Maximization (M) step. In the E-step we compute expected value over latent variable, and in M-step we maximize the likelihood.

(Pages 19-24: Expectation-Maximization for GMM)

• Q27: Define the “responsibility” $\gamma (z_{nk})$ in the context of EM for GMMs.

  • A27: The responsibility $\gamma (z_{nk})$ is the posterior probability that data point $x_n$ was generated by the $k$-th Gaussian component, given the data point: $\gamma (z_{nk})=p(z_{nk}=1|x_n)=\frac{{\pi }_k\mathcal{N}(x_n|{\mu }_k,{\Sigma }_k)}{{\sum }_{j=1}^K{\pi }_j\mathcal{N}(x_n|{\mu }_j,{\Sigma }_j)}$

• Q28: Explain how the responsibilities are calculated using Bayes’ theorem.

  • A28: $\gamma (z_{nk})$ is calculated using Bayes’ rule: $p(z_{nk}=1|x_n)=\frac{p(x_n|z_{nk}=1)p(z_{nk}=1)}{p(x_n)}$ The numerator is $p(x_n|z_{nk}=1)p(z_{nk}=1)=\mathcal{N}(x_n|{\mu }_k,{\Sigma }_k){\pi }_k$. The denominator is $p(x_n)={\sum }_{j=1}^{K}p(x_n|z_{nj}=1)p(z_{nj}=1)={\sum }_{j=1}^K{\pi }_j\mathcal{N}(x_n|{\mu }_j,{\Sigma }_j)$. Therefore, $\gamma (z_{nk})=\frac{{\pi }_k\mathcal{N}(x_n|{\mu }_k,{\Sigma }_k)}{{\sum }_{j=1}^K{\pi }_j\mathcal{N}(x_n|{\mu }_j,{\Sigma }_j)}$.

• Q29: How are the means ${\mu }_k$ updated in the M-step of EM for GMMs?

  • A29: ${\mu }_k^{\text{new}}=\frac{{\sum }_{n=1}^N\gamma (z_{nk})x_n}{{\sum }_{n=1}^N\gamma (z_{nk})}$ This is a weighted average of the data points, where the weights are the responsibilities.

• Q30: How are the covariance matrices ${\Sigma }_k$ updated in the M-step?

  • A30: ${\Sigma }_k^{\text{new}}=\frac{{\sum }_{n=1}^N\gamma (z_{nk})(x_n-{\mu }_k^{\text{new}})(x_n-{\mu }_k^{\text{new}}{)}^T}{{\sum }_{n=1}^N\gamma (z_{nk})}$ This is a weighted average of the outer products of the differences between data points and the updated mean.

• Q31: How are the mixing coefficients ${\pi }_k$ updated in the M-step?

  • A31: ${\pi }_k^{\text{new}}=\frac{1}{N}{\sum }_{n=1}^N\gamma (z_{nk})$ This is the average responsibility for the k-th component over all data points.

(Page 25: Algorithm Summary)

• Q32: Summarize the complete EM algorithm for GMMs.

  • A32:
    1. Initialization: Initialize the means ${\mu }_k$, covariance matrices ${\Sigma }_k$, and mixing coefficients ${\pi }_k$. Choose initial values (can be random or based on K-means).
    2. Compute Initial Log-Likelihood: Calculate the initial log-likelihood: $\log p(X|\pi ,\mu ,\Sigma )$.
    3. E-Step (Expectation): Calculate the responsibilities $\gamma (z_{nk})$ for each data point $x_n$ and each component $k$: $\gamma (z_{nk})=\frac{{\pi }_k\mathcal{N}(x_n|{\mu }_k,{\Sigma }_k)}{{\sum }_{j=1}^K{\pi }_j\mathcal{N}(x_n|{\mu }_j,{\Sigma }_j)}$
    4. M-Step (Maximization): Update the parameters using the responsibilities:
       • ${\mu }_k^{\text{new}}=\frac{{\sum }_{n=1}^N\gamma (z_{nk})x_n}{{\sum }_{n=1}^N\gamma (z_{nk})}$
       • ${\Sigma }_k^{\text{new}}=\frac{{\sum }_{n=1}^N\gamma (z_{nk})(x_n-{\mu }_k^{\text{new}})(x_n-{\mu }_k^{\text{new}}{)}^T}{{\sum }_{n=1}^N\gamma (z_{nk})}$
       • ${\pi }_k^{\text{new}}=\frac{1}{N}{\sum }_{n=1}^N\gamma (z_{nk})$
    5. Compute Log-Likelihood: Calculate the new log-likelihood: $\log p(X|{\pi }^{\text{new}},{\mu }^{\text{new}},{\Sigma }^{\text{new}})$.
    6. Convergence Check: If the log-likelihood has converged (change is below a threshold) or a maximum number of iterations is reached, stop. Otherwise, go back to step 3.

• Q33: What is difference between K-Means and EM Algorithm?

  • A33: K-means performs hard assignments: each data point belongs to exactly one cluster ($r_{nk}$ is either 0 or 1). EM, on the other hand, performs soft assignments: each data point has a probability (responsibility) of belonging to each cluster ($\gamma (z_{nk})$ is a value between 0 and 1). This makes EM more flexible and less sensitive to initialization in some cases. Also, K-means only updates the centroid (means) of cluster, whereas EM algorithm for GMM updates the means, covariance matrices, and mixing coefficients.

This comprehensive Q&A covers all the main points of the lecture slides, emphasizing the theoretical understanding and mathematical formulations required for an exam. Remember to review the derivations and understand the intuition behind each step of the algorithms. Good luck!