I see these terms being used and I keep getting them mixed up. Is there a simple explanation of the differences between them?
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The likelihood function usually depends on many parameters. Depending on the application, we are usually interested in only a subset of these parameters. For example, in linear regression, interest typically lies in the slope coefficients and not on the error variance.
Denote the parameters we are interested in as $\beta$ and the parameters that are not of primary interest as $\theta$. The standard way to approach the estimation problem is to maximize the likelihood function so that we obtain estimates of $\beta$ and $\theta$. However, since the primary interest lies in $\beta$ partial, profile and marginal likelihood offer alternative ways to estimate $\beta$ without estimating $\theta$.
In order to see the difference denote the standard likelihood by $L(\beta, \theta|\mathrm{data})$.
Maximum Likelihood
Find $\beta$ and $\theta$ that maximizes $L(\beta, \theta|\mathrm{data})$.
Partial Likelihood
If we can write the likelihood function as:
$$L(\beta, \theta|\mathrm{data}) = L_1(\beta|\mathrm{data}) L_2(\theta|\mathrm{data})$$
Then we simply maximize $L_1(\beta|\mathrm{data})$.
Profile Likelihood
If we can express $\theta$ as a function of $\beta$ then we replace $\theta$ with the corresponding function.
Say, $\theta = g(\beta)$. Then, we maximize:
$$L(\beta, g(\beta)|\mathrm{data})$$
Marginal Likelihood
We integrate out $\theta$ from the likelihood equation by exploiting the fact that we can identify the probability distribution of $\theta$ conditional on $\beta$.
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2$\begingroup$ Note that the last definition here is an Integrated (or Bayesian) Likelihood, not a Marginal Likelihood. $\endgroup$ – ars Jul 31 '10 at 0:09
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$\begingroup$ Is this correct in the RHS for partial likelihood: "L2(θ|theta)"? $\endgroup$ – jpalecek Jul 31 '10 at 0:13
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$\begingroup$ @ars, would you please edit the answer and provide the definition of Marginal Likelihood then? $\endgroup$ – Waldir Leoncio Oct 3 '16 at 17:06
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$\begingroup$ hmm, the standard way of defining likelihood is the probability of the DATA GIVEN the PARAMETERS, that is, the inverse conditioning of what's written here $\endgroup$ – Andreas Steimer Jul 6 '20 at 21:40
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All three are used when dealing with nuisance parameters in the completely specified likelihood function.
The marginal likelihood is the primary method to eliminate nuisance parameters in theory. It's a true likelihood function (i.e. it's proportional to the (marginal) probability of the observed data).
The partial likelihood is not a true likelihood in general. However, in some cases it can be treated as a likelihood for asymptotic inference. For example in Cox proportional hazards models, where it originated, we're interested in the observed rankings in the data (T1 > T2 > ..) without specifying the baseline hazard. Efron showed that the partial likelihood loses little to no information for a variety of hazard functions.
The profile likelihood is convenient when we have a multidimensional likelihood function and a single parameter of interest. It's specified by replacing the nuisance S by its MLE at each fixed T (the parameter of interest), i.e. L(T) = L(T, S(T)). This can work well in practice, though there is a potential bias in the MLE obtained in this way; the marginal likelihood corrects for this bias.
