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If the magnitude of covariance doesn't really convey any reliable information, and correlation provides us the same information but also gives the degree of dependence, then why use covariance at all ?

warmzombie
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  • Covariance convey the information about the authentic variability while correlation does not. On the other hand, Pearson correlation captures only the strength of linear relation, whereas covariance isn't devoted just to linearity (unless the variabilities are taken under control). You may see my opinion [here](http://stats.stackexchange.com/a/62699/3277). Sometimes we will prefer covariances, sometimes correlations. – ttnphns Dec 13 '13 at 06:32

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I would say that covariance gives you additional information, which is the scale of the input and output standard deviation since $\sigma_{xy}=\sigma_{x}\sigma_{y}\rho_{xy}$ where the terms denote the covariance, the standard deviations, and the correlation, respectively.

On example, where the covariance is needed is modeling data with a multivariate Gaussian distribution $$p(\mathbf x) = \frac{1}{\sqrt{|2\pi\Sigma|}}\exp\left(-\frac{1}{2}(\mathbf x - \mu)^\top \Sigma^{-1}(\mathbf x - \mu)\right).$$ Here, the data covariance is the maximum likelihood estimator of the covariance of the Gaussian. A direct consequence of that is the formula for the conditional mean $$\mathbf x_2 = \Sigma_{21}\Sigma^{-1}_{11}(\mathbf x_1 - \mu_1) + \mu_2$$ where $\mathbf x_1$ and $\mathbf x_2$ denote a partition of the vector $\mathbf x$. The formula basically means that you predict parts of $\mathbf x$ from another part. In case of Gaussian data, this is the optimal linear predictor for least squares error.

fabee
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  • Yes the scaling information bit makes a lot of sense. Sorry I'm still a beginner, didn't quite get the rest of your answer. But thanks for your response! – warmzombie Dec 13 '13 at 08:02
  • You are welcome. If you like the response or if you find it useful, you could upvote or/and accept it. And, of course, welcome to stats.stackexchange. – fabee Dec 13 '13 at 12:31