There is something I am not sure about. If the random variable I'm modelling/predicitng is a 'proportion' (therefore between 0 and 1). So say $X \sim Beta(\alpha, \beta)$.
Now if I want to evaluate my model, do I evaluate it as any other Bayesian regression model (using Standardised Pearson residuals)? So I guess my question is if Standardised Pearson residuals are invariant to the range of the value that we are predicting (if say its between [0,1] or [-100,100])?
I gather that you're trying to predict the measurement $y \in [0,1]$ from some predictor, $x$. For example, predict water proportion as a function of, say, temperature. This is straight forward. One approach is as follows:
First, y is beta distributed around a trend specified by $\alpha$ and $\beta$: $$y_i \sim \mbox{beta}( \alpha_i , \beta_i )$$
Re-parameterize $\alpha$ and $\beta$ in terms of the mode ($\omega$) and concentration ($\kappa$) of the beta distribution: $$ \alpha_i = \omega_i (\kappa-2) + 1 \\ \beta_i = (1-\omega_i) (\kappa-2) + 1 $$ Notice above that $\kappa$ is assumed to be the same for all $i$, which is analogous to the usual assumption of homogeneity of variance.
Then make the mode, $\omega_i$, a logistic function of the predictor, $x_i$: $$ \omega_i = \mbox{logistic}( \beta_0 + \beta_1 x_i ) $$
Finally, put some sensible prior distribution on $\kappa$, $\beta_0$, and $\beta_1$.
In summary, the model says that the central tendency of $y$ is a logistic function of $x$. The central tendency is denoted $\omega$, and it's the mode of the beta-distributed $y$ values at $x$. The lack of noise around the central tendency is the concentration, $\kappa$.
This sort of re-parameterization of the beta distribution is discussed in Chapter 6 of DBDA2E and then used repeatedly in many models throughout the rest of that book. It's straight forward to express the re-parameterization in JAGS. See, for example, p. 239 of DBDA2E.
