The Bayesian Information Criterion (BIC) is proportional to the log of the maximised likelihood. The likelihood is a density with units given by the inverse units of the parameters.
We are free to set the units of the parameters. For example in a linear regression using millilitres (ml) of rainfall to predict height of plants in meters (m), the parameter associated with rainfall has units m/ml.
Therefore, we can also change the units of the likelihood. In this example we could measure rain in litres (l) instead. The units of the likelihood function would change from ml/m to l/m.
The value of the maximum likelihood will depend on the units in which it is specified. Hence, by changing the units of the parameters, we can change the value of the maximum likelihood to be anything we like.
As a result, using the BIC to compare models with different parameters is useless, as all results depend on the units used in each model. Where have I gone wrong? Thank you.