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When reconstructing the ancestral states on a phylogenetic tree given the states at the tips, there are a number of methods for performing the reconstruction. This question is about marginal and joint maximum likelihood reconstruction.

To quote the wiki, marginal reconstrucion "is akin to a greedy algorithm that makes the locally optimal choice at each stage of the optimization problem", while joint reconstruction "find(s) the joint combination of ancestral character states throughout the tree which jointly maximizes the likelihood of the data"

Pupko et al. (2000) describe an algorithm for performing joint reconstruction.

I have a question about a paragraph in the discussion of this paper, where they discuss the differences in the results between the joint and marginal reconstructions.

Deciding which is ‘‘ more correct ’’ depends on the question asked. For instance, if one wishes to count the number of threonine- to-methionine replacements over the entire tree, then the joint reconstruction should be used to obtain this number (2, in our case, on the branch connecting node 24 to node 3 and the branch connecting node 32 to node 33). However, if one wishes to synthesize the hypothetical cytochrome b sequence of the ancestor of Cetartiodactyla, then one should use the marginal reconstruction approach. We emphasize that both methods compute op- timal reconstructions by using all of the available data. Discrepancies originate not from misuse of information, but from the difference in the nature of the probabilistic questions asked

I don't understand why you would use marginal reconstruction for estimating the ancestral state at one node, or how the question you ask affects which method you should use. There is only one true ancestral state, I would think that the correct method for any question is the one that most accurately estimates the true ancestral state. Can someone shed some light on why the marginal reconstruction method is preferable in this situation?

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I know this is too late, but since this is something I'm struggling with too I thought I'd post here in case others also find there way here. I don't have an answer but, here are some quotes that might help (I found them at least partially helpful).

Yang (2004): "Marginal reconstruction is more suitable when one wants the sequence at a particular node, as in the molecular restoration studies. Joint reconstruction is more suitable when one counts changes at each site."

http://aracnologia.macn.gov.ar/st/biblio/Yang%202006%20Computational%20Molecular%20Evolution.pdf

Revell (2014):

"Joint reconstruction is finding the set of character states at all nodes that (jointly) maximizes the likelihood."

"Marginal reconstruction is finding the state at the current node that maximizes the likelihood integrating over all other states at all nodes, in proportion to their probability"

http://www.phytools.org/eqg/Lecture_5.1/Revell.ancestral-state-reconstruction.pdf

Joint reconstruction is "not (necessarily) equivalent to picking the state at each node with the highest probability" (Revell 2014), but is the method of picking the tree with the overall highest MLE.

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