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For a gamma distribution, the answer to this question shows that you can just use the log of the gamma distribution density function.

Is the same true for inverse gamma? It is the same as the log of the IG density?

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The likelihood function given a continuous probability density function $f(.,\theta)$, where $\theta$ is a vector of parameters, and a sample $(x_1,\ldots,x_n)$ is defined as $$ L(x_1,\ldots,x_n,\theta) = \prod_{i=1}^n f(x_i,\theta). $$ The log-likelihood function $l$ is just the natural logarithm of $L$ thus $$ l(x_1,\ldots,x_n,\theta) = \ln L(x_1,\ldots,x_n,\theta) = \sum_{i=1}^n \ln f(x_i,\theta). $$ Of course this is true if $f$ is the pdf of the Gamma distribution or the inverse gamma.

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