Item Response Modeling (IRT) is a technique used by psychologists and cognitive researchers to primarily assess question difficulty and student ability, where ability and difficulty are learned latent factors. Of note, a given {student:question} pair receives a single scalar value, most often binary to represent a correct or incorrect response.

I am interested in a similar model design but with one noteworthy difference: I’d like to model responses to marketing survey data. A given {customer:product} pair can receive one of three class inputs [price, durability, ease-of-use]. In other words, customers submit which factor is most important to them when making a purchase decision regarding a specific product.

As an output, for each customer and each product, a 3D vector should be inferred, where each vector element represents the affiliation with each class, [price, durability, ease-of-use]. (Ex: Some customers might be motivated entirely by price, where others are motivated evenly by all three item characteristics.)

I have the following questions:

- Does such a model design already exist, and if so, can you cite/link the resource?
- What prior/likelihood choices would be appropriate?

In the simplest IRT models, a Bernoulli likelihood can be used: `Bern(ability - difficulty)`

. However, with a 3D vector, perhaps a categorical distribution would be more appropriate: `Cat( cust_[p, d, e ] - prod_[p, d, e] )`

. However, if customers were allowed to denote their affiliation with each latent factor (ex: 1/3 price, 2/3 durability, 0/3 ease of use) then perhaps a Dirichlet distribution would be more appropriate. Or in the event that customers and products were aggregated into groups, perhaps a multinomial likelihood would be most appropriate given a summation of responses.

I list some scenarios above to demonstrate some potential use cases.