Bias-free model fitting of correlated data in interferometry

In optical and infrared long-baseline interferometry, data often display significant correlated errors because of uncertain multiplicative factors such as the instrumental transfer function or the pixel-to-visibility matrix. In the context of model fitting, this situation often leads to a significant bias in the model parameters. In the most severe cases this can can result in a fit lying outside of the range of measurement values. This is known in nuclear physics as Peelle's Pertinent Puzzle. I show how this arises in the context of interferometry and determine that the relative bias is of the order of the square root of the correlated component of the relative uncertainty times the number of measurements. It impacts preferentially large data sets, such as those obtained in medium to high spectral resolution. I then give a conceptually simple and computationally cheap way to avoid the issue: model the data without covariances, estimate the covariance matrix by error propagation using the modelled data instead of the actual data, and perform the model fitting using the covariance matrix. I also show that a more imprecise but also unbiased result can be obtained from ignoring correlations in the model fitting.