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Param_mapping
Parameter mappingsBy default, the model assumes all unknown variables (except precision hyperparameters) are continuous variables that can vary between -Inf and Inf. However, one may want to constrain the range of values of some of these variables. For example, in the context of a Q-learning model:
as being dummy variables that are passed through an appropriate mapping , i.e.:
This mapping can now be inserted into the evolution and observation functions, so that although the VB algorithm derives an approximate posterior Note: The distribution
The quality of this approximation strongly depends on how linear is the mapping. See here for better approximations in the context of the exponential and sigmoidal mappings. We will now see examples of useful one-to-one mappings below: Sigmoidal mappingThis mapping is relevant when restricting the variables to the unit [0,1] interval.
Note: by rescaling and translating the above sigmoid mapping, one can construct variables that are bounded from above and from below with any arbitrary values:
where the effective parameter Exponential mappingThis mapping is relevant when restricting the variables to positive numbers:
Note: the lower bound of the constraint can be changed by translating the above mapping... | |
as being dummy variables
that are passed through an appropriate mapping
, i.e.:
on the unbounded variable
fulfills the desired constraints.
on the effective parameter can be approximated using a first-order Taylor expansion of the mapping, as follows: