{
    name: "Glenfiddich 15",
    notes: [
        {
            user_name: "Derick",
            date: ISODate("2013-10-02"),
            note: "Caramel and honey, fruity with some hints of earth",
            likes: 17,
            rating: 3,
        }
    ]
}
<?php
$whisky = new (\MongoDB\Client)->demo->whisky;

$whisky->updateOne(
    [ 'name' => 'Glenfiddich 15', 'notes.user_name' => 'Derick' ],
    [ '$inc => [ 'notes.$.likes' => 1 ] ]
);
?>