Regression for an outcome (ratio or fraction) between 0 and 1 I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a > 0$ and $b > 0$. So, the ratio would be between $0$ and $1$.
I could use linear regression, although it doesn't naturally limit to 0..1.  I have no reason to believe the relationship is linear, but of course it is often used anyway, as a simple first model.
I could use a logistic regression, although it is normally used to predict the probability of a two-state outcome, not to predict a continuous value from the range 0..1.
Knowing nothing more, would you use linear regression, logistic regression, or hidden option c?
 A: You should choose "hidden option c", where c is beta regression.  This is a type of regression model that is appropriate when the response variable is distributed as Beta.  You can think of it as analogous to a generalized linear model.  It's exactly what you are looking for.  There is a package in R called betareg which deals with this.  I don't know if you use R, but even if you don't you could read the 'vignettes' anyway, they will give you general information about the topic in addition to how to implement it in R (which you wouldn't need in that case).  

Edit (much later):  Let me make a quick clarification.  I interpret the question as being about the ratio of two, positive, real values.  If so, (and they are distributed as Gammas) that is a Beta distribution.  However, if $a$ is a count of 'successes' out of a known total, $b$, of 'trials', then this would be a count proportion $a/b$, not a continuous proportion, and you should use binomial GLM (e.g., logistic regression). For how to do it in R, see e.g. How to do logistic regression in R when outcome is fractional (a ratio of two counts)?
Another possibility is to use linear regression if the ratios can be transformed so as to meet the assumptions of a standard linear model, although I would not be optimistic about that actually working.  
A: Are these paired samples or two independent populations?
If independent populations, you might consider log(M) = log(B) + $X_i$*log(ratio).  M is your measurement (a vector containing all values of A and B) and X is a vector $X_i$ = 1 if $M_i$ is a value of A, $X_i$ = 0 if $M_i$ is a value of B. 
Your intercept of this regression will be log(B) and your slope will be log(ratio).
See more here:
Beyene J, Moineddin R. Methods for confidence interval estimation of a ratio parameter with application to location quotients. BMC medical research methodology. 2005;5(1):32.
EDIT: I have written an SPSS addon to do just this. I can share it if you're interested.
A: Not true.  The data for logistic regression is binary 0 or 1 but the model predicts p say the probability of success given the predictors $X_i$, $i=1,2,..,k$ where $k$ is the number of predictor variables in the model.  Actually because of the logit function the linear model predicts the value of log($\frac{p}{1-p}$).  So to get the prediction for p you just do the inverse transformation $p=\frac{\exp(x)}{[1+\exp(x)]}$ where $x$ is the predicted logit.
A: We can use sample_weights in SVM-C or any other classifier with weights being the ratio. There would be two data points for each data point in the original case:


*

*With 1 as target variable where sample_weight is equal to ratio

*With 0 as target variable where sample_weight is (1-ratio).


Consider the effect of sample_weights here on SVM-C https://scikit-learn.org/stable/auto_examples/svm/plot_weighted_samples.html 
We can then use the predicted probability for target value 1 as our ratio estimate. Here weights refer to the sample_weights in the fit method of sklearn
