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In probability, we can find the cdf using the pdf and vise-versa. Integrating pdf yields the cdf.

Does integrating the likelihood function yield any important thing?

In statistics, $\mathcal{L} (M\mid X )= P(X \mid M) $.

If, cdf is to pdf, then likelihood is to what?

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Likelihood integration (with respect to $M$) doesn't make sense because it doesn't even integrate/sum to $1$ if we iterate over all possible $M$'s. However, integration/sum of likelihood multiplied with prior makes sense because the integral is equal to the evidence.

With respect to $x$, it's still a density/PMF and its integral is a CDF.

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