What type of regression model do I use? $y = \mathbf{X} \beta$ + $\epsilon$ + $m$
Where y, $\epsilon$, and $m$ are $n \times 1$ column vectors, $\beta$ is $p \times 1$ and $\mathbf{X}$ is $n \times p$. 
$y$ is a noisy time-series signal modeled by linear combinations of a basis set $\mathbf{X}$ discribed by $\beta$.  $\epsilon$ is additive Gaussian noise.  $m$ is an estimate of the mismatch inherent in the reconstruction.  The columns of $\textbf{X}$ can't perfectly reconstruct $y$, even if it were noiseless.  Am I correct in my understanding that this can be called an errors-in-variables model?
So given y and $\textbf{X}$, what is $\beta$?
Here's the twist:  The true value of $\beta$ should either be 0 or 1, while all other variables are comprised of continuous data.  What I'd like to be able to do is find $\beta$ to be a value between 0 or 1 which corresponds to a probability the corresponding column of $\textbf{X}$ is present in the reconstruction.  A value in $\beta$ of 1 indicates that its corresponding column of x is present, 0 not. 
Since the columns of $\textbf{X}$ can't perfectly reconstruct $y$ even if it were noiseless, basic methods like OLS give answers where elements of $\beta$ are above 1 and below 0, which for me are unphysical.  
In other words, is there a way to constrain the results of a linear regression ($y = \mathbf{X} \beta + \epsilon + m$) to only admit $\beta$ values between zero and 1 in a way which they can be interpreted as a probability that $\beta$ equals one or not?  Perhaps through the use of a prior distribution?
If anyone has any keywords I can look into, or references to point out, that would be very appreciated!
P.S. If anyone can give a Bayesian interpretation.. Bonus points!  The people this is for love that stuff.
Thanks in advance!
 A: You can always force the $\beta$s to be between 0 and 1 by rewriting the likelihood in terms of $\beta_j = f(\beta^*)$ for some function $f$, e.g. $1/(1+\exp(-\beta^*))$. Then optimise the $\beta^*$s but report the $\beta$s.  However that won't automatically get you an interpretation in terms of the probability that $\beta$ is relevant to the regression.
So I think the first thing to do is  separate, conceptually speaking, the question of the probability that each $\beta_j$ is equal to zero i.e. not included/relevant from the value it should take if it is relevant.
As for models, the keywords are probably Bayesian and  automatic relevance determination.  Assume a Bayesian model with a hierarchical prior such that $p(\beta \mid \alpha) = \prod_j N(0, \alpha_j^{-1})$ and maybe a gamma prior on the $\alpha$s if you're going to sample.  As $\alpha_j$ goes to infinity, $\beta_j$ is more probably equal to zero.
Then integrate out (or optimise) the (ir)relevance parameters $\alpha$ during posterior inference to get predictions for $y$ and separately examine the $\alpha$s to give you an idea which $\beta$s are irrelevant.  For more details on these methods, the references I have to hand are MacKay (ms) or Tipping's work, and a tech report by Minka et al. discussing fitting stategies.  
I've no idea what is your $m$ parameter(s), so I may have missed something to do with that.
A: If you use simple regression (say OLS), then $\beta$ can be converted to a t-stat, which then can be converted to a p-value, which ranges between $0$ and $1$; a p-value of $0$ for $\beta_i$ means a significant contribution of $X_i$ to $y$, whereas a p-value of 1 means no contribution of $X_i$ to $y$.
Does this help?
A: If you don't know anything about t-stats and p-values then you really need the help of a professional statitician on this project, or take some stats courses (what you are asking is beyond the intro courses) until you have enough knowledge to do this.
Along the way as you are learning (or discussing with a consultant), least squares regression with bounds on the slopes is just a case of quadradic programming and many stats and math programs have routines for doing quadradic programming.
If you want to do this (assuming "this" is well defined) using Bayesian methods then it is pretty straight forward, just choose priors that limit your parameters to the range desired (uniform or beta priors, possibly discrete priors).
You may also want to look at model averaging or Bayesian model averaging, these can give answers of the probability of an x variable being a predictor of the y variable.
We need to understand the problem better (and you need to understand the statistical modeling better) in order to give more specific advice.
A: No. This is not an errors in variables problem. That would refer to noise in the INDEPENDENT variables, i.e., X.
Your problem is a bound constrained OLS problem, where the variables are constrained to lie in the closed interval [0,1]. Can you truly interpret these numbers as probabilities? That seems a stretch, if you use a least squares in any form to do the estimation. The numbers are then simply linear combinations of the independent variables, NOT probabilities.
For the numbers to make sense as probabilities, you would arguably need to do some variety of maximum likelihood estimation. Of course, you can always feel free to interpret a number in any way you want, but if you want a meaningful result, it is best to use statistics properly. For that to work, you need to clearly define what it means for a column to be part of the model. A linear combination is NOT a probability.
