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Bayesian statistics
The Bayesian philosophy involves a completely different approach to statistics.
The Bayesian version of estimation is considered for the basic situation concerning the estimation of a parameter given a random sample from a particular distribution. Classical estimation involves the method of maximum likelihood.
The fundamental difference between Bayesian and classical methods is that the parameter θ is considered to be a random variable in Bayesian methods.
In classical statistics θ is a fixed but unknown quantity. This leads to difficulties such as the careful interpretation required for classical confidence intervals, where it is the interval that is random. As soon as the data are observed and a numerical interval is calculated, there is no probability involved. A statement such as P(10.45 <θ < 13.26)= 0.95 cannot be made because θ is not a random variable.
In Bayesian statistics no such difficulties arise and probability statements can be made concerning the values of a parameter θ .
This means that it is quite possible to calculate a Bayesian confidence interval for a parameter.
Another advantage of Bayesian statistics is that it enables us to make use of any information that we already have about the situation under investigation. Often researchers investigating an unknown population parameter have information available
from other sources in advance of the study that provides a strong indication of what values the parameter is likely to take. This additional information might be in a form that cannot be incorporated directly in the current study. The classical statistical
approach offers no scope for the researchers to take this additional information into account. However, the Bayesian approach does allow additional information to be taken into account when trying to estimate a population parameter.
The Bayesian version of estimation is considered for the basic situation concerning the estimation of a parameter given a random sample from a particular distribution. Classical estimation involves the method of maximum likelihood.
The fundamental difference between Bayesian and classical methods is that the parameter θ is considered to be a random variable in Bayesian methods.
In classical statistics θ is a fixed but unknown quantity. This leads to difficulties such as the careful interpretation required for classical confidence intervals, where it is the interval that is random. As soon as the data are observed and a numerical interval is calculated, there is no probability involved. A statement such as P(10.45 <θ < 13.26)= 0.95 cannot be made because θ is not a random variable.
In Bayesian statistics no such difficulties arise and probability statements can be made concerning the values of a parameter θ .
This means that it is quite possible to calculate a Bayesian confidence interval for a parameter.
Another advantage of Bayesian statistics is that it enables us to make use of any information that we already have about the situation under investigation. Often researchers investigating an unknown population parameter have information available
from other sources in advance of the study that provides a strong indication of what values the parameter is likely to take. This additional information might be in a form that cannot be incorporated directly in the current study. The classical statistical
approach offers no scope for the researchers to take this additional information into account. However, the Bayesian approach does allow additional information to be taken into account when trying to estimate a population parameter.
Tags: Bayes, Quants, Statistics, theorem
Views: 198
Replies to This Discussion
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Permalink Reply by Shashank Shekhar on -
If anyone wants any asistance in Bayesian Stats, Discrete Maths, Game Theory, Number Theory or other Quantitative methods, they can mail me back or leave a comment on my comment Wall. I will be more than happy to respond.
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Permalink Reply by Arun on -
Oh! I've got questions on all of those topics... I think I'll need to hire you around for sometime.. ;)
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Permalink Reply by Arun on
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I'm quite not able to make out the difference from this. Two questions that could help me would be:
1. Why would one choose between Bayesian and Classical Statistics? The pros and cons of both.
2. Under what circumstances would one outweigh the other?
I think if you could highlight these two points, it would be very helpful...
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Permalink Reply by Shashank Shekhar on
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As i said the Bayesian Stats allow us to take the additional information into account unlike classical stats. An example of this would be where an insurance company is reviewing its premium rates for a particular type of policy and has access to results from other insurers, as well as from its own policyholders. This information cannot be taken into account directly because the terms and conditions of the policies for other companies may be slightly different. However, these additional data might contain a lot of useful information, which should not be ignored. In classical statistics we cant use the aditional information because of difference in nature of data.
Bayesian Statistics should be always preferred over classical statistics and especially in cases where we need to compare Hypotheses.
Am also attaching a file for your reference. Please let me know if this helps. - Attachments:
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Permalink Reply by Vincent Granville on -
Classical statistics with penalized likelihoods is (in my opinion) the same as Bayesian statistics.
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Permalink Reply by Gambo on -
Dear Mr.Shekhar,
Just came back from the field (conducting survey) the topic you raised is very interesting. Just printed the attachment, will surely get back to you once am through. Its nice to have a different view on the same issue. Up here in Nigeria, most of us, Statisticians are of the classical school, one feels like a salmon swimming aginst the tide when you talk about bayesian approaches to some issues.
See you soon.
Tanks
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Permalink Reply by Shashank Shekhar on
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Sure Mr. Gambo. I would love to discuss the intricacies of Bayesian approach.
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