Dealing with 0,1 values in a beta regression I have some data in [0,1] which I would like to analyze with a beta regression.
Of course something needs to be done to accommodate the 0,1 values.  I dislike
modifying data to fit a model.  also I don't believe that  zero and 1 inflation
is a good idea because I believe in this case one should  consider the 0's
to be very small positive values  (but I don't want to say exactly what
value is appropriate.  A reasonable choice I believe would be to pick small values
like  .001 and .999  and to fit the model using the cumulative dist for the beta.
So for observations y_i the log likelihood LL_iwould be
 if  y_i < .001   LL+=log(cumd_beta(.001))
 else if y_i>.999  LL+=log(1.0-cum_beta(.999))
 else LL+=log(beta_density(y_i))

What I like about this model is that if the beta regression model is valid
this model is also valid, but it removes a bit of the sensitivity to the
extreme values. However this seems to be such a natural approach that
I wonder why I don't find any obvious references in the literature.
So my question is instead of modifying the data, why not modify the model.
Modifying the data biases the results (based on the assumption that the original model is valid), whereas modifying the model by binnning the extreme values does not bias the results.
Maybe there is a problem I am overlooking?
 A: I think the actual "correct" answer to this question is zero-one inflated beta regression. This is designed to handle data that vary continuously on the interval [0,1], and allows many real 0's and 1's to be in the data. This approach fits three separate models in a bayesian context, similar to what @B_Miner proposed.
Model 1: Is a value a discrete 0/1, or is the value in (0,1)? Fit with a bernoulli distribution.
Model 2: Fit discrete subset with a bernoulli distribution. 
Model 3: Fit (0,1) subset with beta regression. 
For prediction, the first model results can be used to weight the predictions of models 2 and 3. This can be implemented within the zoib R package, or home-brewed in BUGS/JAGS/STAN/etc. 
A: According to Smithson & Verkuilen (2006)$^1$, an appropriate transformation is
$$ x' = \frac{x(N-1) + s}{N} $$

"where N is the sample size and s is a constant between 0 and 1. From a Bayesian standpoint, s acts as if we are taking a prior into account. A reasonable choice for s would be .5."

This will squeeze data that lies in $[0,1]$ to be in $(0,1)$. The above quote, and a mathematical reason of the transformation is available in the [paper's supplementary notes].

Reference:

*

*Smithson, M. & Verkuilen, J. A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol. Methods 11, 54–71 (2006). DOI: 10.1037/1082-989X.11.1.54
A: Dave,
A common approach to this problem is to fit 2 logistic regression models to predict whether a case is 0 or 1. Then, a beta regression is used for those in the range (0,1).
A: The beta distribution follows from the sufficient statistics $(\log(x), \log(1-x))$.  Do those statistics make sense for your data?  If you have so many zeros and ones, then it seems doubtful that they do, and you might consider not using a beta distribution at all.
If you were to choose the sufficient statistic $x$ instead (over your bounded support), then I believe you end up with a truncated exponential distribution, and with $(x,x^2)$ a truncated normal distribution.
I believe that both are easily estimated in a Bayesian way as they are both exponential families.  This is a modification of the model as you were hoping.
A: My experience with regression modeling, in general, involves also the exercise of judgment and in that regard, it is something of an art, especially when it comes to constructing good forecasting models. My observation is that actually parsimonious models appear to be superior, overfitting a model is not the apparent optimal best approach. My experience includes time series modeling and reading of the works of Box-Jenkins both of which expound judgment based parsimonious modeling.
If one reads the original work on beta regression, “BETA REGRESSION FOR MODELLING RATES AND PROPORTIONS”, the authors suggest several link functions or the use of no link function. The idea that the odds ratio link function is the most appropriate, or the only choice, given its singularity at O and 1, with many such (or near such) points in the data, is just bad judgment in my view.
I just noticed comments by Neil, "Do those statistics make sense for your data? If you have so many zeros and ones, then it seems doubtful that they do.." which appears to echo my point.
