# Discrete choice models; estimating probability of new alternatives

Let's say the set of alternatives are $j=1,2,...J$, where $X_i$ is the vector of subject-specific attributes for subject $i$, and $Z_{ij}$ is the vector of alternative-specific attributes for alternative $j$ available to subject $i$. If a new alternative $J^*$ is introduced, how can I estimate the probability of $J^*$ being selected by a given subject $i$, and update the probabilities of existing alternatives?

I have two different examples in mind:

1) current alternatives are bus, plane, car, and $J^*$ (the new alternative) will be train. I want to estimate the probability that a subject i (who has certain age and income) selects each of these now four alternatives (now including train), for a given price and time for each alternative.

2) alternatives are different neighborhoods. A new neighborhood (which wasn't developed initially) is now developed. I wan to estimate the probability that a household i, with certain size, income and number of children under 18, chooses this new neighborhood which has certain attributes (average rent, amenity score, school score, safety score). Note that in contrast to the previous example, attributes of an alternative in this case are constant across different observations.

It will be perhaps a separate question and may need a separate post, but in this second example, the number of alternatives can be very large (a few thousand), and I will need to use random subsets for estimating the model parameters.

• Could you describe the data you have? – whuber Nov 22 '16 at 17:49
• @whuber I am trying to design my research question, and at this point I don't have real data, but edited the question to include two examples of how my data structure would look like. The latter will probably closer to what I will need. – Reza Amindarbari Nov 23 '16 at 20:27