The mind is what the brain does. Obesity can be a choice in the same way that going through with cryonics can be a choice. As a materialist, I see little difference; both are the outcome of many physical processes, some of which run through the brain, which typically can themselves can be traced causally further back. (For example, I have little doubt that were it possible to run such a study, we would find a high heritability to cryonics as well as the already-established obesity/BMI heritabilities, because cryonics seems to relate very heavily to various cognitive attitudes which are closely connected to other cognitive traits which are heritable, such as intuitive religious cognition which tends towards dualism or essentialism and against the reductionism & materialism which leads to patternism.)

For the purposes of this argument, it's sufficient that merely some fraction of people can indeed choose to stop becoming obese, which does indeed appear to be the case.