*(Sandipan Dey **21 August 2016)*

- In this article, the
*Gaussian Conjugate Priors*will be used to compute the*Posterior distribution*for some online dataset (1D and 2D) following*Gaussian Distribution*. - First, a N(μ0,σ0)
*prior*is chosen to model the unknown mean μ variable of the data, while assuming the data variance σ2 as fixed. - Then a few
*i.i.d.*samples are drawn from a Gaussian distribution the prior belief about the mean μ is updated, X_i∼N(μ,σ2) with*prior*μ∼N(μ0,σ0). - The
*posterior*probability distribution is also a*Gaussian*distribution as shown in the figure below from the videos of professor Herbert Lee from the*coursera course Bayesian Statistics Concepts by UCSC*. - Then the
*recursive Bayesian updates*and the*prior*and*posterior hyper-parameters*and the means are updated as and when a new datapoint is received. Also, the*frequentist’s MLE*and*95% confidence interval*are computed, along with the Bayesian*95% credible interval*. - The following animation shows the results of simulation of 50 such data samples, starting with the prior N(0,10).
- The left bottom plot visualizes the
*histogram*of the data generated. - Every time a new datapoint is received, the
*prior*belief is updated. - The right bottom table represents the summary statistics.
*Prior*and*Posterior*means (of the arrival rate) respectively correspond to the previous and updated beliefs about the mean of the data. - The next animation shows the same results (with contours) modeling a set of 2D
*Gaussian*samples.

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