No magic trick I fear - I just thought hard about what can be learned from the data and what cannot, did quite a bit of math and then tested a bunch of things until I found the one that worked. Also simulating data and seeing how changes in the parameters change the shape of the curve helped me get some insight.
If I get it right, t is known while A,k,d are paraemters, right? So treating Y = f(A, k, d, t) One way to go about this might be to take take t_{min}, t_{max} as the min/max t you observe and try to use y_{min} = f(A, k, d, t_{min}) and y_{max} = f(A, k, d, t_{max}) as new parameters - if you can solve for the other parameters given y_{min}, y_{max} - but I am really just guessing here. The value at midpoint of the observed t range might also be a good parameter. The point is that such values are by construction well constrained by data (but might be impractical because it is hard to derive the parameters you need from them).
EDIT: To be a bit more specific, since Gompertz is also a sigmoid curve. Here are some specific ideas I used with the logistic sigmoid:
- If I observe only the start of the curve, the upper plateau (A) is not determined.
- So instead we use the value at midpoint of observed data as a parameter. When I observe the upper plateau, this would roughly correspond to A/2, but it is constrained by data even when I only see the lower plateau.
- If the inflection point of the curve is far from the observed data range, I only see the lower or the upper plateau, i.e. almost constant function. In other words if the inflection point is far from observed data, it has little influence on the actual shape of the curve. In this case the “slope” of the curve is also not informed by the data. To overcome this, we used the location of the inflection point on the x-axis as another parameter AND we put an informative prior on it to constrain it “close” to the range of observed data. This makes the slope somewhat identified as well.
- This changes the interpretation of the model fit! If the posterior for the inflection point has notable mass outside the observed data range, we have to be aware that this part of the posterior is influenced almost exclusively by the prior and the actual inflection point can be much further from the observed data range than what the posterior might suggest. In this case the fitted slope is also just a consequence of the prior.
For the logistic sigmoid those resulted in reasonably neat formulae, not sure if that would be the case for Gompertz, so other tricks might be necessary.