I have a discrete probability distribution which I want to update as new evidence comes in. I thought about it in a Bayesian context, but I don't think that will work because I don't have any formulation of the likelihood. Perhaps I'm missing something… Here's the problem statement:
Suppose I am running an online store, and people can order stuff online and choose a time slot for it to be delivered.Assume I have three delivery time slots [6:00-8:00, 8:00-10:00, 10:00-12:00]. My goal is to get a real-time estimate for the amount of orders I will receive at the end of the day for each slot.
More formally, let the three slots be denoted $[x_1,x_2,x_3]$. I am interested in knowing the values of orders ($y$) in each of the slots $[y_1,y_2,y_3]$ at the end of the day. So predicting number of orders in each time slot.
I was thinking: if I assume an initial distribution (normalized) of orders $[p_1, p_2, p_3]$, I want to update this distribution over time as new orders are coming in. We here assume the total number of orders to remain constant. More practically:
I have initial probability distribution $[p_1, p_2, p_3] = [0.4, 0.4, 0.2]$ (based on domain knowledge, people tend to order less later in the day for some reason)
After 10 orders have come in, almost everyone ordered in the first time slot. $y_10 = [9, 1, 0]$
I want to update my prior to reflect this. Perhaps it would be something like $[0.5, 0.4, 0.1]$ based on the evidence that came in
Repeat throughout the day. Because I assume the total number of orders to be constant, this would give me a direct estimate of the orders $[y_1,y_2,y_3]$.
If I knew the likelihood function I could use some kind of discrete Bayesian updating with prior distribution being $[p_1, p_2, p_3]$ and update as orders come in. The problem is I don't know the likelihood function. How could I achieve this?