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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 not 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.

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    $\begingroup$ In a multivariate setting, the Bayesian Gaussian update is still simple: you just sum the inverse-variances $\endgroup$ – Guillaume Dehaene Nov 1 '16 at 10:38
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This sounds to me like an application of the Kalman Filter:

Let x^t be a vector defining the true product quality of n products. x^b is the perceived product quality of the same n products. y defines an observation of a particular product's quality.

Before the consumer purchases a new product, we have prior information about the errors in our estimate of product quality. This is described by B = E[(x^b-x^t)(x^b-x^t)^T]. We also have prior knowledge of the errors in the observation of a particular product's quality, R = E[(y-Hx^t)(y-Hx^t)^T]. Here, H is a matrix that maps the state to the observation space. If the consumer just purchased the first product in x, H is [1 0 0 ... 0].

After purchasing a product, we update our state estimate via x^a = x^b + K(y-Hx^b), where K = BH^t(HBH^t+R)^(-1) and x^a is our posterior estimate of product quality. Finally, our posterior estimate of the errors in our estimate of product quality is A=B(I-KH), where I is the identity.

https://en.wikipedia.org/wiki/Kalman_filter

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