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The following is an excerpt from the book "Algorithms to Live By" by Brian Christian. (p. 230)

...; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view.

I understand what the phrase means, but I'm not so clear about the grammatical structure.

Is "be" verb being omitted as in "those which [he thinks] (are) likeliest to..."? I assume "he thinks" is a mere inserted phrase, so "be likely to" should be the main verb.

My second-guessing goes "think likely." Can you use "think likely" in a similar way to "is/seem/appear likely"?

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    ...Those which he thinks are the most likely to...
    – Kate Bunting
    Jan 27 '21 at 9:06
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    You are correct in saying that to be has been omitted. "He has to pick... those which, he thinks, are [the] likeliest to..." As you say, he thinks is a parenthesis, rather than thinks likely being a phrasal verb.
    – Kate Bunting
    Jan 27 '21 at 14:51
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Your second guess is correct: You can think something likely/pretty/..., just like you can find it likely etc.

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