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Bayesian Point Estimation

What are the Bayesian Point of estimation and what are the process of inference in Bayesian statistics?

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Bayesian Point Estimation:

A) Bayesian Statistics is one way of incorporating prior information about a parameter into the estimation process.

B) Adherents claim that this helps to make the estimation more relevant to the scienti c problem at hand.

C) Opponents counter that it makes statistical inference subjective.

D) The underlying principle of Bayesian statistics also di ers from the more common Frequentist inference that we have covered to date.

E) In Bayesian statistics, all unknown quantities are considered random variables.

F) Thus the parameters of a distribution are now considered random.

G) The usual model is then considered to be a conditional distribution of the data given the parameters.

H) Since the parameter vector θ is considered random it also has a distribution.

I) The marginal distribution of θ is called the Prior Distribution.

J) The prior distribution is supposed to capture our beliefs about θ before the collection of data.

The process of inference in Bayesian statistics is as follows.

1. Specify a conditional distribution of the data given the parameters. This is identical to the usual model speci cation in frequentist statistics.

2. Specify the prior distribution of the model parameters Π(θ).

3. Collect the data, X = x.

4. Update the prior distribution based on the data observed to give a Posterior Distribution of the parameters given the observed data x, Π(θ|x).

5. All inference is then based on this posterior distribution.

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