# Log-likelihood of Normal Distribution: Why the term $frac{n}{2}log(2pi sigma^2)$ is not considered in the minimization of SSE?

Cross Validated Asked by Javier TG on November 29, 2020

Reading some books and papers like the great one: ”Bundle Adjustment – A Modern Synthesis” (page 10), I found that the cost function weigthed Sum of Squared Error (SSE):

$$SSE = frac{1}{2} sum_i Delta z_i(x)^T,W_i,Delta z_i(x)$$ $$,,,,,,,,,$$(respecting the notation from the article linked above)

Represents as well the negative log-likelihood of the Normal Distribution from where the ground-truth data was obtained (considering that $$W_i$$ aproximates the inverse of the covariance matrix). Thereby, minimizing $$SSE$$, we will obtaine the parameters $$x$$ that best fit this Normal Distribution.

However, looking at some posts like this one form Wikipedia, they state that the log-likelihood for the Normal Distribution is given by:

$$log(mathcal{L}(mu,sigma))= -frac{n}{2},log(2pisigma^2)-frac{1}{2sigma^2}sum_{i=1}^n(x_i-mu)^2$$

So, Why the term $$frac{n}{2},log(2pisigma^2)$$ is not considered in the previous reasoning of minimizing $$SSE$$ = maximizing the likelihood?

Because that part of the log likelihood is constant (with respect to $$mu$$). Leaving it out saves some computation, but does not affect the ML estimate.

If you are also estimating $$sigma$$ then you would need to include that part as well.

Correct answer by Greg Snow on November 29, 2020