Closed form solution of multivariate Gaussian over mixture of multivariate Gaussians Suppose I have three variables $X_{1}, X_{2}$ and $Y$, where $X_{1}, X_{2}$ are continuous and $Y$ is binary. The conditional distribution of $X_{1}, X_{2}$, given $Y$ is a multivariate Gaussian distribution so that $P(X_{1}, X_{0}\mid Y=y)\sim \mathcal{N}(\mu_{y},\Sigma)$, with $y\in\{0,1\}$. Notice that the covariance matrix is the same for both values of $Y$. Suppose I also have the distribution of $Y$, but because $Y$ is binary, this distribution can be summarized with one parameter $p:=P(Y=1)$.
Now using Bayes' rule I want to find the conditional distribution of $Y$ given $X_{0}$ and $X_{1}$:
\begin{align}
  P(Y=1\mid \mathbf{X}=\mathbf{x}) &= {P(\mathbf{X}=\mathbf{x}\mid Y=1)P(Y=1) \over P(\mathbf{X}=\mathbf{x})}\\
  &= {p\exp(-\frac{1}{2}(\mathbf{x}-\mu_{1})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{1})) 
  \over 
  p\exp(-\frac{1}{2}(\mathbf{x}-\mu_{1})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{1})) + 
  (1-p)\exp(-\frac{1}{2}(\mathbf{x}-\mu_{0})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{0}))}
\end{align}
Notice that the marginal distribution of $\mathbf{X}$ is a Gaussian mixture. Is there a cleaner expression for this conditional distribution? I have been playing around with it but I don't seem to get anywhere.

$${p\exp(-\frac{1}{2}(\mathbf{x}-\mu_{1})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{1})) 
  \over 
  p\exp(-\frac{1}{2}(\mathbf{x}-\mu_{1})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{1})) + 
  (1-p)\exp(-\frac{1}{2}(\mathbf{x}-\mu_{0})^{\top}\Sigma^{-1}(\mathbf{x}-\mu_{0}))}$$
can be simplified into
$${p\exp(\mathbf x^\top\Sigma^{-1}\mu_1-\frac{1}{2}\mu_{1}^{\top}\Sigma^{-1}\mu_{1}) 
  \over 
  p\exp(\mathbf x^\top\Sigma^{-1}\mu_1-\frac{1}{2}\mu_{1}^{\top}\Sigma^{-1}\mu_{1}) + 
  (1-p)\exp(\mathbf x^\top\Sigma^{-1}\mu_0-\frac{1}{2}\mu_{0}^{\top}\Sigma^{-1}\mu_{0})}$$
or
$$1\Big/1+\frac{1-p}{p}\exp\left(\mathbf x^\top\Sigma^{-1}(\mu_0-\mu_1)-\frac{1}{2}\mu_{0}^{\top}\Sigma^{-1}\mu_{0}+\frac{1}{2}\mu_{1}^{\top}\Sigma^{-1}\mu_{1}\right) 
  $$
