BIC and AIC by hand

Topic: Model selection for stochastic models · more questions on this topic
‹ Posterior odds The Laplace integral in one dimension ›

With n=300, the two-component fit has \widehat\ell=-1100 and d=11, and the three-component fit has \widehat\ell=-1050 and d=17. Compute BIC and AIC for each and state which model each criterion selects.

Worked answer (click to show)

Use one sign convention for every candidate. Here smaller is better. Then compare the improvement in fit with the extra parameter charge. With \log 300=5.7038,

\begin{align*} \mathrm{BIC}(H=2)&=2200+11\times 5.7038=2262.7,\\ \mathrm{AIC}(H=2)&=2200+22=2222,\\ \mathrm{BIC}(H=3)&=2100+17\times 5.7038=2197.0,\\ \mathrm{AIC}(H=3)&=2100+34=2134. \end{align*}

Both criteria select H=3. The gain of 100 in -2\ell more than pays for the six extra parameters under either penalty: 6\times 5.7038=34.2 for BIC and 12 for AIC.