Confidence interval of an average coefficient calculated from curve fits, with each curve fit using a parameter which has its own confidence interval

Question

I am curve fitting a model to data sets in order to determine a coefficient for each curve fit/data set. I then calculate an average coefficient and the 95% confidence interval for this average. However, each curve fit involves the use of a parameter (in the model equation) which itself has a 95% confidence interval. How do I propagate the 95% confidence interval for this parameter into the 95% confidence interval for the coefficient calculated from curve fits?

Here is an example version of the model that I am curve fitting to data: $$y=\text{erfc}\left(\frac{x}{A+B}\right)$$ $A$ is the curve fit variable and $B$ is the pre-determined parameter with a 95% confidence interval. Changing the value of $B$ changes the calculated value of $A$, so unless I'm mistaken, the confidence interval of $B$ should propagate into that of $A$. How is this done mathematically?

Edit: I found the answer to my question. It's included below.

Answer 1

I found a way to accomplish the propagation of uncertainty: Combine $A$ and $B$ into a single term—say, $C$—and curve fit to determine $C$. Then, solve $C = A + B$ for $A$ and calculate the uncertainty of $A$ with the uncertainties of $C$ and $B$.

Answer 2

Your question is a bit confusing -- if x is the adjustable parameter to be optimised, and K is an input (from a disribution), what is the independent variable? Nevertheless, if I assume for the sake of argument that it's there somewhere, conceptually you should fit by optimisation of a single parameter

$y = \text{erfc}(\theta)$,

then rearrange the basic error propagation (eq 3.20 from Bevington) to extract the uncertainty in $x$ (i.e. solve for $\sigma_x$ from the following):

$\sigma_\theta^2 = \sigma_x^2 + \sigma_K^2 \pm 2\sigma_{xK}^2$

where the correlation coefficient of the covariance term would be 1.

This assumes a normal distribution for all quantities, of course, which may not be the case. Monte Carlo sampling would be the more robust way of solving this, if you've got any computational nous.

Answer 3

This is a good question, and unfortunately there is no easy answer. The right way to go about problems like this is via Bayesian inference. In this context, that amounts to assuming that every uncertain variable has a distribution, conditional or unconditional, and then deriving the distribution over what you consider interesting, given any data at hand, and marginalizing (i.e. integrating) over the other variables you consider uninteresting. My advice is to try to state the whole Bayesian model, and then make simplifying assumptions (possibly drastic) to arrive at something tractable.

There are probably many non-Bayesian methods, which amount to specific simplifications of the Bayesian approach, making use of particular assumptions, possibly unstated. My advice is, don't resort immediately to simplifications, instead work out the whole model and consider what are reasonable and workable assumptions for your problem.