I have a nagging question regarding the Normal distribution and maintaining proportionality in Bayesian Inference. Say for example that:
$\pi(\theta|Y) \propto L(Y|\theta)\pi(\theta)$
$Y | \theta \sim N(\theta, \sigma^2)$
$\theta \sim N(\mu, \tau^2)$
If we want to calculate the posterior distribution for $\theta$, we can drop any terms that do not include the parameter of interest (i.e. normalizing constant). This results in the following:
$\pi(\theta|Y) \propto \exp(-\frac{1}{2 \sigma^2}\sum_{i = 1}^{n} (Y_i - \theta)^2) \exp(-\frac{1}{2 \tau^2}(\theta - \mu)^2)$
My question is why is it that we do not drop the $\frac{1}{2}$ in the exponential terms? Why would this no longer be proportional to the distribution? Or do we avoid dropping it for another reason?
Thank you for your help!