I am new to statistics and Bayesian analysis. Therefore, I have some problems that would like to clarify.
Suppose my problem is to calculate the posterior distribution for the time of ship to spend in port. Because there is no historical distribution regarding how long a ship stays in that port, therefore I assumed a uniform distribution [1, 15], with 1 hour denotes the lower bound, and 15 hours denotes the upper bound. This range includes the basic ship operation and cargo handling in port e.g., a ship could arrive and finish cargo handling within 15 hours. Then, I collected some actual ship arrival samples. I collected 50 ship samples that represent the current situation of port time for a ship. Based on the samples, I understand they follow a normal distribution [15, 5] with mean 15 hours, and standard deviation 5 hours.
Base on the information above, I would like to confirm if my understanding on Bayesian is correct:
- Think of the time as observations coming from some distribution, for example, Gaussian with some parameters mu and sigma. If mu and sigma are known, then we can always sample observations from that distribution, and there is no need to perform statistical analysis. However, the values of mu and sigma are usually unknown, and we perform statistical analysis to estimate them. In the frequentist approach, in which parameters mu and sigma are just points, we can approximate them using the sample mean and sample variance. In the Bayesian paradigm, things are a little bit different. Here we assume that mu and sigma have prior distribution on their own. So one of the goals of Bayesian is to estimate the posterior distribution of mu and sigma given the evidence, which is sample data in this case. Generally, we do not estimate the posterior distribution of data or time in the example. We estimate the posterior of the parameters of the distribution that generates this data.
- If my above understanding is correct, I am not sure how to set the prior for the mu and sigma. Given I have the prior uniform distribution [1, 15], can I set the prior distribution of mu to follow the uniform distribution [1, 15], then set a Jeffrey prior for the variance? Or, from your opinion, what prior should I use for variance?
Thank you very much for your kind help. I am really lost about the next step.