## Chapter 4: Exercises

1. (*)AM-GM inequality:
1. Give the formula to show the inequality of arithmetic and geometric means. When does the equality hold?
2. Prove the formula.
2. (*)MLE for an unfair coin: Toss an unfair coin 10 times and obtain 7 heads and 3 tails. Use the principle of MLE (maximum likelihood estimate) to derive the probabilities for head and tail are 0.7 and 0.3, respectively.
3. (*)MLE for a 3-sided dice: Toss a 3-sided dice 10 times and obtain 2 times for side 1, 3 times for side 2, 5 times for side 3. Use the principle of MLE (maximum likelihood estimate) to derive the probabilities for sides 1, 2, and 3 are 0.2, 0.3, and 0.5, respectively.
4. (*)1D Gaussian PDF and its MLE:
1. What is the formula for 1D Gaussian PDF (probability density function)?
2. Given a set of observation $\{x_1, x_2, ..., x_n\}$, what are the MLE (maximum likelihood estimate) of $\mu$ and $\sigma^2$ for the 1D Gaussian PDF?
5. (**)Multivariate Gaussian PDF and its MLE:
1. What is the formula for the multivariate Gaussian PDF (probability density function) in the d-dimensional space?
2. Given a set of observation $\{\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_n\}$, what are the MLE (maximum likelihood estimate) of $\mathbf{\mu} \in R^d$ and $\Sigma \in R^{d \times d}$ for the multivariate Gaussian PDF?
6. (*)Function for computing the MLE of the multivariate Gaussian PDF: The multivariate Gaussiaon PDF in the d-dimensional space is given by $$g(\mathbf{x}, \mathbf{\mu}, \Sigma)= \frac{1}{\sqrt{(2\pi)^d|\Sigma|}} e^{-(\mathbf{x-\mu})^T\Sigma^{-1}(\mathbf{x-\mu})/2}.$$ Write a function mle4gaussian.m with the following usage: [mu, sigma]=mle4gaussian(X); where $X=\left[\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_n \right]$ is a matrix with each column being an observation vector, and mu and sigma are the MLE for the multivariate Gaussian expressed by $$\left\{ \begin{array}{l} \hat{\mathbf{\mu}}=\frac{1}{n}\sum_{i=1}^n\mathbf{x}_i,\\ \hat{\Sigma}=\frac{1}{n}\sum_{i=1}^n \left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right)\left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right)^T.\\ \end{array} \right.$$ Note that you are not allowed to use any of the ready-to-use functions from any MATLAB toolbox. In other words, you need to write the function from scratch.

Here is a test case. If the input X is given by

X=[magic(5), magic(5)']; Then the returned mu is 13 13 13 13 13 And the returned sigma is 52.0000 4.5000 -28.0000 -23.0000 -5.5000 4.5000 47.0000 -3.0000 -25.5000 -23.0000 -28.0000 -3.0000 62.0000 -3.0000 -28.0000 -23.0000 -25.5000 -3.0000 47.0000 4.5000 -5.5000 -23.0000 -28.0000 4.5000 52.0000
7. (*)Simplified formula of the MLE of ND Gaussian PDF: Prove that the MLE of $\Sigma$ of the multivariate Gaussian PDF can be expressed as follows: $$\hat{\Sigma}=\frac{1}{n}\sum_{i=1}^n \left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right)\left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right)^T = \frac{1}{n}XX^T-\hat{\mathbf{\mu}}\hat{\mathbf{\mu}}^T,$$ where $X=\left[\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_n \right]$ is a matrix with each column being an observation vector.
8. (*)Incremental formula for the MLE of ND Gaussian PDF: Given the observation matrix $X=\left[\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_n \right]$ where each column is an observation vector, the MLE of a D-dim Gaussian PDF can be expressed as: $$\left\{ \begin{array}{l} \hat{\mathbf{\mu}}=\frac{1}{n}\sum_{i=1}^n\mathbf{x}_i,\\ \hat{\Sigma}=\frac{1}{n}XX^T-\hat{\mathbf{\mu}}\hat{\mathbf{\mu}}^T.\\ \end{array} \right.$$ We can use $\hat{\mathbf{\mu}}_{-k}$ and $\hat{\Sigma}_{-k}$ to denote the MLE after removing $\mathbf{x}_k$ from $X$, Prove that the MLE can be expressed as: $$\left\{ \begin{array}{l} \hat{\mathbf{\mu}}_{-k}=\frac{n\hat{\mathbf{\mu}}-\mathbf{x}_k}{n-1},\\ \hat{\Sigma}_{-k}=\frac{n}{n-1}\hat{\Sigma}+\frac{n}{n-1}\hat{\mathbf{\mu}}\hat{\mathbf{\mu}}^T-\hat{\mathbf{\mu}}_{-k}\hat{\mathbf{\mu}}_{-k}^T -\frac{1}{n-1}\mathbf{x}_k\mathbf{x}_k^T.\\ \end{array} \right.$$
9. (*)ND Gaussian PDF with not-full covariance matrix: Given a set of observation $\{\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_n\}$, we want to find the MLE of $\mathbf{\mu}$ and $\Sigma$ for the multivariate Gaussian PDF when $\Sigma$ assumes a specific non-full forms.
1. If $\Sigma$ is restricted to be diagonal, that is $$\Sigma=diag(\alpha_1, \cdots, \alpha_d)= \left[ \begin{array}{ccc} \alpha_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \alpha_d \\ \end{array} \right]$$ Prove that the MLE of $\mathbf{\mu}$ and $\Sigma$ are $$\left\{ \begin{array}{l} \hat{\mathbf{\mu}}=\frac{1}{n}\sum_{i=1}^n\mathbf{x}_i,\\ \hat{\alpha}_k=\frac{1}{n}\sum_{i=1}^n \left(\mathbf{x}_i(k)-\hat{\mathbf{\mu}}(k)\right)^2, k=1, \ldots, d.\\ \end{array} \right.$$
2. If $\Sigma$ is restricted to be $\alpha I$ (a constant times an identity matrix), prove that the MLE of $\mathbf{\mu}$ and $\Sigma$ are $$\left\{ \begin{array}{l} \hat{\mathbf{\mu}}=\frac{1}{n}\sum_{i=1}^n\mathbf{x}_i,\\ \hat{\alpha}=\frac{1}{nd}\sum_{i=1}^n \left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right)^T\left(\mathbf{x}_i-\hat{\mathbf{\mu}}\right).\\ \end{array} \right.$$

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