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BIC is most often calculated by maximizing the log likelihood function. However, it is also possible to calculate BIC with residual sums of squares. This is pretty easy to find online and not an issue for me. However, what is odd and a bit confusing is all of the variants of BIC calculated with RSS/SSE that I have seen online. To go into more detail, here are four of the different versions I have seen:

On this document it appears as...

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On this document it appears as...

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On this Stack Overflow question it appears as...

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And on Wikipedia it appears as...

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I'm left a bit confused here and with numerous questions. Most broadly, what is the correct implementation or variant of BIC to use using RSS?

More specifically: (1) Is -2 * log necessary? The second referenced document states, "Note that the term -2 * ln L for used in this specialization is equal to the rescaled normal loglikelihood up to an additive constant that depends only on n." Frankly, I am not sure what this means. Are they saying that adding -2 * ln to this formula rescales/normalizes the log likelihood (but it isn't a log likelihood...)? (2) Is the error variance term used in some of the BIC formulas (but not all) necessary?

Any help here would be appreciated. I feel like I am just throwing darts in terms of which one of these to use.

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  • $\begingroup$ Your 2nd and 4th look the same to me $\endgroup$
    – Glen_b
    Commented Aug 23, 2019 at 4:38
  • $\begingroup$ Oops, you're right about that. Sorry about that. Do you think the others variants should probably be dismissed then in favor of that one? $\endgroup$
    – JElder
    Commented Aug 23, 2019 at 4:58
  • $\begingroup$ Rather, some clarification and something of a frame challenge is in order (i.e. I reject some premises on which the question relies, like that a version using RSS is necessarily distinct from one that maximizes likelihood). In many cases it won't matter in the least which one you use (as long as you're consistent), in others it can -- but you don't say what you're using it for (nor what sort of model you're using it on), so we can't even judge if it matters for you. What would make a cirterion "the right one"? Schwarz' original definition? Some other particular definition? $\endgroup$
    – Glen_b
    Commented Aug 23, 2019 at 5:19
  • $\begingroup$ Thanks for following up. I am using BIC for model comparison and RSS/maximum likelihood for parameter estimation and model fitting on trial-by-trial data in a reinforcement learning paradigm (psychology/neuroscience type of reinforcement learning). You basically take the dataset (consisting of choices and rewards), a set of free parameters to estimate, and loop over each subject fitting the model by minimizing -2LL (or in this context RSS). Then BIC is used to compare models with different parameters. BIC is preferred/normative as it penalizes more for more parameters than AIC. $\endgroup$
    – JElder
    Commented Aug 23, 2019 at 16:29

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