Model Settings Sample Clauses

Model Settings. Basic Parameters and Ter- minology To model the non-Markovian stochastic process of respondents’ decision making, we first use the binary random variable Bi to denote the final decision of the ith respondent in the survey with the subscript ”M ” for ”Markovian” and ”NM ” for ”non-Markovian”. Let Bi in both cases satisfy the following:   1 if the respondent decides to buy Bi =  0 if the respondent decides not to buy Then, the result of the survey will form a chain filled with ”1” and ”0”. Assume all respondents’ purchase probabilities in the Markov process at a given price follow an independent identical binary distribution1. Then   1 with probability P0 Bi,M =  0 with probability 1 — P0 1This is also referred to as respondents’ innate purchase probability. By maximum likelihood estimation(see e.g. (30)), we have: P0 = ⟨ N ΣN Bi,M ⟩ If we examine the definition of P0 more carefully, we’ll find it shares the same meaning as the sample’s final purchase rate. As shown in the last chapter, the final purchase rate in each sample converges, with each value read from data as follow: ,  ,, ▇▇(▇▇,▇ = 1|price = A10) = 0.322 , P0(Bi,M = 1|price = A5) = 0.548 0 i,M , P (B = 1|price = A3) = 0.781 The non-Markovian purchase probability Pi for the ith respondent should also depend on decisions made by previous Nr respondents which form his/her mem- ory. Assume the decision of each respondent in the memory has the same weight 1/Nr; then the overall effect on the last respondent should relates to the purchase percentage ri in the memory; by using previous notations, it can be defined as the following: 3.1.1. Let ri be the purchase percentage in the memory which is re- ceived by the he ith respondent. ri = ΣNr j=1 Bi−j,NM (3.1) To keep consistent with the survey, we set B1,NM = B1,M and the superscript on Σ changes to i — 1 for the case i ≤ Nr. In the rest of this chapter, we’ll propose several models which can provide us with Pi if given P0 and ri.
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