# Derive the joint probability density function of differences of Gaussian Mixtures

Consider a 3-variate random vector $$(\epsilon_0, \epsilon_1, \epsilon_2)$$ which is distributed as a Gaussian mixture: (with some abuse of notation) $$f(\epsilon_0, \epsilon_1, \epsilon_2)=\underbrace{w_a \mathcal{N}\Big(\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, \begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}\Big)}_{\text{component (b)}}$$ where $$f$$ denotes the PDF, $$(w_a, w_b)$$ are the mixture weights, $$\mathcal{N}(\mu, \Sigma)$$ stays for 3-variate normal distribution with mean $$\mu$$ and var-cov matrix $$\Sigma$$.

Is it possible to derive $$f(\epsilon_1-\epsilon_0, \epsilon_1-\epsilon_2)$$? If yes, could you walk me through the derivation?

I'm confused on how to proceed:

• Deriving the (marginal) PDF of $$(\epsilon_1-\epsilon_0)$$ and $$(\epsilon_1-\epsilon_2)$$ when $$\{\epsilon_0, \epsilon_1, \epsilon_2\}$$ are mutually independent is explained here for example.

• Here, however, I don't have mutual independence. Moreover, I want the joint of pairs of differences.

Any hint would be extremely appreciated.

• Sorry when I try to answer you question it was mixed up. Could you try to recover to your original question? – user158565 Nov 1 '18 at 20:55

Let $$\left (\begin{matrix} \epsilon_1-\epsilon_0 \\ \epsilon_1-\epsilon_2\end{matrix}\right) = \left (\begin{matrix} 1 & -1 & 0 \\ 1& 0 & -1 \end{matrix}\right)\left (\begin{matrix} \epsilon_0 \\ \epsilon_1\\\epsilon_2\end{matrix}\right) = A\epsilon$$
Then $$f(\epsilon_0-\epsilon_1, \epsilon_0-\epsilon_2)=\underbrace{w_a \mathcal{N}\Big(A\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, A\begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}A'\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(A\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, A\begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}A'\Big)}_{\text{component (b)}}$$
• is $A'$ inside the brackets? – user3285148 Nov 1 '18 at 21:13
• Thanks. 1) I think to have $A$ as you define the vector of epsilons should be ordered $(\epsilon_1, \epsilon_0, \epsilon_2)$. 2) I'm confused about the way you work with mixtures: according to your methodology, for example, $f(\epsilon_1-\epsilon_0)=w_a \mathcal{N}(0, 2(\sigma^2_a-\rho_a))+w_b \mathcal{N}(0, 2(\sigma^2_b-\rho_b))$. – user3285148 Nov 2 '18 at 9:45