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Could anyone recommend a good similarity measure for objects which have multiple classes, where each class is part of a hierarchy?

For example, let's say the classes look like:

1 Produce
  1.1 Eggs
    1.1.1 Duck eggs
    1.1.2 Chicken eggs
  1.2 Milk
    1.2.1 Cow milk
    1.2.2 Goat milk
2 Baked goods
  2.1 Cakes
    2.1.1 Cheesecake
    2.1.2 Chocolate

An object might be tagged with items from the above at any level, e.g.:

Omelette: eggs, milk (1.1, 1.2)
Duck egg omelette: duck eggs, milk (1.1.1, 1.2)
Goat milk chocolate cheesecake: goat milk, cheesecake, chocolate (1.2.2, 2.1.1, 2.1.2)
Beef: produce (1)

If the classes weren't part of a hierarchy, I'd probably I'd look at cosine similarity (or equivalent) between classes assigned to an object, but I'd like to use the fact that different classes with the same parents also have some similarity value (e.g. in the example above, beef has some small similarity to omelette, since they both have items from the class '1 produce').

If it helps, the hierarchy has ~200k classes, with a maximum depth of 5.

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While I don't have enough expertise to advise you on selection of the best similarity measure, I've seen a number of them in various papers. The following collection of research papers hopefully will be useful to you in determining the optimal measure for your research. Please note that I intentionally included papers, using both frequentist and Bayesian approaches to hierarchical classification, including class information, for the sake of more comprehensive coverage.

Frequentist approach:

Bayesian approach:

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    $\begingroup$ Thanks for those links, turns out the 2nd one down was almost exactly what I was after. $\endgroup$ – Dave Challis Jan 9 '15 at 10:23
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    $\begingroup$ @DaveChallis: My pleasure! Glad to be able to help. $\endgroup$ – Aleksandr Blekh Jan 9 '15 at 11:24

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