Body: <p>Let be a Bayesian multivariate normal distribution classifier with distinct covariance matrices for each class and isotropic, i.e. with equal values over the entire diagonal and zero otherwise, <span class="math-container">$\mathbf{\Sigma}_i=\sigma_i^2\mathbf{I},~\forall i$</span>.</p>
  <p>How can I compute the equation for estimating the parameter <span class="math-container">$\sigma_{i}$</span> by the maximum likelihood method? Here <span class="math-container">$\sigma_{i,j}$</span> is is the covariance between <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span>. So <span class="math-container">$\sigma_i$</span> is just the variance of <span class="math-container">$x_i$</span>.</p>
  <p><strong>Attempt:</strong></p>
  <p>Suppose <span class="math-container">$\mathcal{X}_i = \{x^t_i\}^N_{t=1}$</span> i.i.d, <span class="math-container">$x_i^t$</span> is in the class <span class="math-container">$C_i$</span> and <span class="math-container">$x_i^t \sim \mathcal{N}(\mu, \sigma^2)$</span>.</p>
  <p>Do I have to find the log-likelihood under <span class="math-container">$p(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[ -\frac{(x-\mu)^2}{2 \sigma^2}\right]$</span>, find the derivative and put it equal to <span class="math-container">$0$</span> to find the maximum?</p>
  <p><strong>EDIT</strong></p>
  <p>Suppose my data points are <span class="math-container">$m$</span>-dimensional, and I have <span class="math-container">$K$</span> classes.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-09-25T20:40:22.877Z
Comments.Text: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/113411/discussion-on-question-by-david-estimating-sigma-i-according-to-maximum-likel).
Comments.UserDisplayName: nbro
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Comments.Text: How can your data points be $m$ dimensional if they follow a univariate normal distribution?
Comments.UserDisplayName: David
Tags: <classification><probability><probability-distribution><maximum-likelihood>
Title: Estimating $\sigma_i$ according to maximum likelihood method
UserDisplayName: David
ThreadText: # Estimating $\sigma_i$ according to maximum likelihood method
  
  (Asked by: username_0 on 2020-09-25)
  
  Let be a Bayesian multivariate normal distribution classifier with distinct covariance matrices for each class and isotropic, i.e. with equal values over the entire diagonal and zero otherwise, $\mathbf{\Sigma}\_i=\sigma\_i^2\mathbf{I},~\forall i$.
  
  How can I compute the equation for estimating the parameter $\sigma\_{i}$ by the maximum likelihood method? Here $\sigma\_{i,j}$ is is the covariance between $x\_i$ and $x\_j$. So $\sigma\_i$ is just the variance of $x\_i$.
  
  **Attempt:**
  
  Suppose $\mathcal{X}\_i = \{x^t\_i\}^N\_{t=1}$ i.i.d, $x\_i^t$ is in the class $C\_i$ and $x\_i^t \sim \mathcal{N}(\mu, \sigma^2)$.
  
  Do I have to find the log-likelihood under $p(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[ -\frac{(x-\mu)^2}{2 \sigma^2}\right]$, find the derivative and put it equal to $0$ to find the maximum?
  
  **EDIT**
  
  Suppose my data points are $m$-dimensional, and I have $K$ classes.
  
  **Tags:** classification, probability, probability-distribution, maximum-likelihood
  > **Comments:**
  > **username_1**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/113411/discussion-on-question-by-david-estimating-sigma-i-according-to-maximum-likel).  
  > **username_0**: How can your data points be $m$ dimensional if they follow a univariate normal distribution?  
  
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  ## 0 answers
  
  