The algorithm used in the retrieval of geophysical quantities from the Atmospheric Infrared Sounder (AIRS) instrument depends on two fundamental components. The first is a cost function that is the sum of squares of the differences between cloud-cleared radiances and their corresponding forward-model terms. The second is the minimization of this cost function. For the retrieval of carbon dioxide, the minimization is further improved using the method of vanishing partial derivatives (VPDs). In this letter, we show that this VPD component is identical to a coordinate descent method with Newton-Raphson updates, which allows it to be put in context with other optimization algorithms. We also show that the AIRS cost function is a limiting case of the cost function used in optimal estimation, which demonstrates how uncertainty quantification in the AIRS retrieval can be implemented.