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"Machine Learning: A Probabilistic Perspective by Kevin Patrick Murphy" in page 31 says

iff there exist function g and h such that

p(x, y|z) = g(x, z)h(y, z) ...

I guess exist here should be exists, is it?

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    The book actually says "iff there exist functions g and h such that…"
    – Ben Kovitz
    Oct 8, 2019 at 0:35

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Both forms are possible.

If there exist..... is the subjunctive form of the verb (about which you will find hundreds of illustrations on the net).

It's like saying:If there be another challenger, let him come forward. - a construction that has a somewhat poetic or theatrical ring.

There's also the question of whether you could interpret function g and h as a single function or a double function (although a single is indicated).

In practice, most people would probably use exists, as English speakers make ever less use of the subjunctive.

https://en.bab.la/conjugation/english/exist

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  • Two problems: 1. "If there exist…" is indicative, agreeing with a plural subject in the conventional way; details here. 2. "function g and h" is a transcription error. That book has "functions g and h".
    – Ben Kovitz
    Oct 8, 2019 at 0:46