Any general material on how to update with multivariate normal priors would do., such as text book chatpers or paper, notes etc. I tried to google, nothing similar/relevant was found.
I read this in a paper(Erdem 1998) but could understand how they derive the posterior upto to time T. No idea how to type equations here so i used a screen shot for my questions.
Some background information. The key assumption here is consumers does have a perfect perceptions of the TRUE quality of a product (brand quality etc.). But the true product quality is fixed in perspective of the company. The consumer’s perceived quality is different from true quality but depend on the true quality. Each time they purchase, they update their belief on the true quality (or mean quality). The consumers evaluate the product quality as a Bayesian.
There are several papers dealing with the Bayesian theoretical framework needed for the prediction of how consumers will behave over time, and how their perceptions and evaluations will change according to models based on multivariable normal priors. I hope that the following readings could be useful:
Yalcinkaya G, Aktekin T. Brand Extension Effects and Core Attributes of Experience Product Franchises: A Bayesian Approach. Journal of Product Innovation Management 2015; 32: 731-746.
Zeithammer R, Lenk P. Bayesian estimation of multivariate-normal models when dimensions are absent. Quant Market Econ 2006; 4: 241.
Rust TR, InmanJJ, Jia J, Zahorik A. What You DON'T Know About Customer-Perceived Quality: The Role of Customer Expectation Distributions. Marketing Science 1999; 18: 77-92.