# BIC useless as it depends on units?

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.

• One of the prerequisites for direct use of information criteria for model comparison is that the dependent variable is exactly the same between the different models. Different units make it different. – Richard Hardy Feb 20 '18 at 14:08
• A slight aside: I thought a probability density was a unitless quantity as it is the same as a "density" in physics or calculus, an infinitesimal quantity. I do not think this fact relates to whether common units implies BIC is appropriate for comparing models with different parameters; it is probably neither a sufficient or necessary condition. – AdamO Feb 20 '18 at 15:03
• What do you mean by "changing the units of parameters"? Do you mean regressors in a multivariate model? Most inference, including the likelihood ratio, is invariant to changes of scale in these parameters. – AdamO Feb 20 '18 at 15:05
• Thanks for your comments. @RichardHardy even if we keep the dependent variable in the same units, wont the units of the likelihood function still depend on the units of the independent variables? The independent variables (and their units) will vary across models. – Josh12 Feb 20 '18 at 17:06
• @AdamO isn't it the case that if we integrate the likelihood function across all parameter space we will end up with a probability which has no units? Therefore the original density that was integrated over must have units given by the inverse of the units of the parameters. When I say change units of the parameters I mean changing the units of the predictors, yep. In a regression this would change the units of the associated parameters. Why would the likelihood ratio be invariant to this? We can construct exactly the same argument used against the BIC for the likelihood ratio. Thank you! – Josh12 Feb 20 '18 at 17:09