(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 prior is chosen to model the unknown mean variable of the data, while assuming the data variance as fixed.
- Then a few i.i.d. samples are drawn from a Gaussian distribution the prior belief about the mean is updated, with prior .
- 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 .
- 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.