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Is the posterior distribution always a compromise between prior and likelihood?
The posterior distribution is always a compromise between the prior distribution and the likelihood function. The previous chapter (specifically Section 5.3) gave examples by using grid approximation, but now we can illustrate the compromise with a mathematical formula.
How does the likelihood affect the posterior shape?
In the first and second plot the likelihood is nearly invisible because the posterior sits right on top of it. When the prior has only 1 or 2 data points worth of information, it has essentially no impact on the posterior shape (footnote 2).
When is the posterior and prior in the same family?
If the prior and the posterior distribution are in the same family,the prior and posterior are called conjugatedistributions. The beta distribution is a conjugate prior because the posterior is also a beta distribution. We say that the beta distribution is the conjugate family for the binomial likelihood.
How to find the posterior distribution of left handed students?
Find the posterior distribution of left-handed students in the population assuming uninformative prior. Summarize the results. According to the literature 5-20% of people are left-handed. Take this information into account in your prior and calculate new posterior. I know the beta distribution should be used here. First, with α and β values as 1?
How is the likelihood function different from the probability function?
By contrast, the likelihood function is continuous because the probability parameter p can take on any of the infinite values between 0 and 1. The probabilities in the top plot sum to 1, whereas the integral of the continuous likelihood function in the bottom panel is much less than 1; that is, the likelihoods do not sum to 1.
Which is the posterior term in probability and likelihood?
The likelihood term, P(Y|X) is the probability of getting a result for a given value of the parameters. It is what you label probability. The posterior and prior terms are what you describe as likelihoods.
What’s the difference between probability and likelihood in Bayes?
The probabilities in the top plot sum to 1, whereas the integral of the continuous likelihood function in the bottom panel is much less than 1; that is, the likelihoods do not sum to 1. The difference between probability and likelihood becomes clear when one uses the probability distribution function in general-purpose programming languages.