# Newton-Raphson method to solve for dof when performing MLE of a multivariate Student-t distribution using EM

I am reading the derivation of EM algorithm to estimate the maximum likelihood of a multivariate Student-t distribution $$\mathcal{T}(\mathbf{x} \vert \pmb{\mu}, \pmb{\Sigma}, \nu)$$ in Kevin Murphy's textbook (Machine Learning: a probabilistic perspective, section 11.4.5). I can follow the whole section, but have a concern at the very last step when determining the dof $$\nu$$.

The E-step for $$\nu$$ is: $$$$\mathbb{E}\left[L_{G}(\nu) \right] = -N \log \Gamma \left(\frac{\nu}{2} \right) + \frac{N\nu}{2} \log \left(\frac{\nu}{2}\right) + \frac{\nu}{2} \sum_{i=1}^{N} \left( \bar{\ell}^{(t)}_{i} - \bar{z}_{i}^{(t)} \right), \tag{11.77} \label{11.77}$$$$ where $$\Gamma(x)$$ is the gamma function.

The M-step is to maximize $$\mathbb{E}\left[L_{G}(\nu) \right]$$ w.r.t. $$\nu$$ by solving for the root of the first derivative: $$$$\frac{\partial \mathbb{E}\left[L_{G}(\nu) \right]}{\partial \nu} = -\frac{N}{2} \Psi \left(\frac{\nu}{2} \right) + \frac{N}{2} \log \left(\frac{\nu}{2} \right) + \frac{N}{2} + \frac{1}{2}\sum_{i=1}^{N} \left( \bar{\ell}^{(t)}_{i} - \bar{z}_{i}^{(t)} \right), \tag{11.78} \label{11.78}$$$$ where $$\Psi(x)$$ is the digamma function.

As discussed in the textbook, the root of \eqref{11.78} can be found by using some numerical methods for root finding, such as half-interval method. However, I wonder if we can apply Newton-Raphson method for \eqref{11.78} by taking second derivative: $$$$\frac{\partial^2 \mathbb{E}\left[L_{G}(\nu) \right]}{\partial\nu^{2}} = -\frac{N}{4} \Psi^{(2)} \left(\frac{\nu}{2} \right) + \frac{N}{2\nu},$$$$ where $$\Psi^{(2)}(x)$$ is the trigamma function.

Given the second derivative, we can apply the Newton-Raphson method to iteratively find the dof $$\nu$$ as: $$$$\nu \gets \nu - \left(\frac{\partial \mathbb{E}\left[L_{G}(\nu) \right]}{\partial \nu}\right)/\left(\frac{\partial^2 \mathbb{E}\left[L_{G}(\nu) \right]}{\partial\nu^{2}}\right).$$$$

My question is if we can apply Newton-Raphson method to find $$\nu$$. What are the disadvantages of the Newton-Raphson method preventing us from using in this case? I am aware that Newton-Raphson method requires the second derivative, but in this case, the second derivative can be efficiently calculated. Or, I might be wrong because of the complexity of the digamma calculation.