# Are Jacobian adjustments necessary when the target parameter is a difference between two parameters?

**URL:** <https://discourse.mc-stan.org/t/are-jacobian-adjustments-necessary-when-the-target-parameter-is-a-difference-between-two-parameters/15918>\
**Category:** Modeling\
**Tags:** jacobian-adjustment, specification\
**Created:** [June 13, 2020, 4:10pm UTC](https://discourse.mc-stan.org/t/are-jacobian-adjustments-necessary-when-the-target-parameter-is-a-difference-between-two-parameters/15918 "2020-06-13T16:10:37Z")\
**Posts on this page:** 1\
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**Author:** ![betanalpha](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/betanalpha/32/4_2.png) [@betanalpha](https://discourse.mc-stan.org/u/betanalpha)\
**Post date:** [June 13, 2020, 9:03pm UTC](https://discourse.mc-stan.org/t/are-jacobian-adjustments-necessary-when-the-target-parameter-is-a-difference-between-two-parameters/15918/4 "2020-06-13T21:03:50Z")

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A quick warning – Jacobians are defined for only 1-1 functions. There is not a well-defined Jacobian for a 2-1 map that takes in two parameters and returns a single parameter. Formally you have to map \theta\_{1} and \theta\_{2} to _two_ parameters, say \delta = \theta\_{1} - \theta\_{2} and \sigma = \theta\_{1} + \theta\_{2}, compute the Jacobian (which here would be a constant and hence ignorable), and then marginalize out the second variable \sigma.

Regarding the question of bias – in general there is no guarantee that a Bayesian posterior, or any posterior expectation value such as a posterior mean, will be close to the true value. The figures you showed don’t seem to indicate any particularly pathological behavior.

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