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When do you use a posterior normal distribution?
When a prior dataset can be roughly represented by a normal distribution, bayesian statistics show that sample information from the same process can be used to obtain a posterior normal distribution. The latter is a weighted combination of the prior and the sample.
Which is the normal distribution with mean and variance?
Thus, conditional on and , is normal with mean and variance . Conditional on , the prior predictive distribution of is where is an vector of ones, and is the identity matrix. This can be derived from the case where is known (see above).
What’s the difference between normal distribution and Bayesian estimation?
The main difference is that we need to replace the prior mean with the posterior mean and the prior variance with the posterior variance . As in the previous section, the sample is assumed to be a vector of IID draws from a normal distribution. However, we now assume that not only the mean , but also the variance is unknown.
Which is the best guess of the prior distribution?
It may be clear that, if there is no local information we bring to bear, the best guess is the prior distribution which then can not be updated. Posterior distribution, the revised, or “updated” prior, based on the sample of new information (3). Note that this distribution is that of the mean.
How does the posterior approach the estimated value?
If you plot the results, you’ll see how posterior approaches the estimated value (it’s true value is marked by red line) as new data is accumulated. For learning more you can check those slides and Conjugate Bayesian analysis of the Gaussian distribution paper by Kevin P. Murphy.
How is the posterior predictive distribution of a vector calculated?
The posterior predictive distribution Assume that new observations are drawn independently from the same normal distribution from which have been extracted. The posterior predictive distribution of the vector is where is the identity matrix and is a vector of ones.
How is a prior distribution constructed in Bayesian update?
The latter is a weighted combination of the prior and the sample. The larger the sample and the smaller the sample variance, the higher the weight that the sample information receives. A prior distribution can be constructed by collecting data, or by “subjective experience” which can not be formally processed.