I have an array of data that is normally distributed, i.e. we're dealing with a multivariate Gaussian. We write the data as $X = \{x_1, x_2, \ldots , x_N\}$

So, there are unknown parameters $\mu$ and $\Sigma$

Using a subset of data, I would like to write down a likelihood for the variance of this subset of data, i.e. $\operatorname{Var}(\{x_1,\ldots, x_7\})$.

I can then measure the actual data points, and then compute their empirical variance.

Question: How do you write down a likelihood for the variance? I'm confused how to do this in terms of the variance.

My guess if you write down the log-likelihood function as a function of the variance, and then plot it against a series of values.


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