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> such as it's extremely hard to calculate priors. I think you mean it is difficult to formulate priors? Typically, calculating a prior only involves sampling

by arcanus 11y ago

> such as it's extremely hard to calculate priors. I think you mean it is difficult to formulate priors? Typically, calculating a prior only involves sampling from a distribution.

Fomite 11y ago
Though technically, obtaining that distribution, if you're doing anything above "Eh, it's probably normal..." could be described as "hard".
eanzenberg 11y ago
Let's use this example: http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/ http://betterexplained.com/articles/an-intuitive-and-short-e... -1% of women have breast cancer (and therefore 99% do not). -80% of mammograms detect breast cancer when it is there (and therefore 20% miss it). -9.6% of mammograms detect breast cancer when it’s not there (and therefore 90.4% correctly return a negative result). For generating a prior, I posit it's really hard to determine that 1% of women have breast cancer, and this prior is almost linearly sensitive to whether you actually have cancer: prior | % likelihood of cancer 2% | 16% 1% | 7.8% 0.5% | 4% Why do I think it's hard to determine the prior? Imagine you are a 34 yrs old woman in SF. Do you use the rate of breast cancer for only 34 yr olds? For the range of 30-40 yr olds? 30-65 yr olds? For women only in SF? In the bay area? In California? Data from 2000-2016? 2010-2016? This becomes hard because positive cases are rare (thank god), so the population of which you calculate the prior from can be noisy. Imagine if the positive result occurs 1e-5 times.. you are talking about 10's of cases every million. Vary the prior probability and your posterior probability will vary just as much.
- arcanus 11y ago
  hmm, what you are discussing above is typically called the formulation of the prior, not the calculation of it. And I agree that this formulation can be problematic. It is almost certainly the most contentious element in Bayesian inference. However, I will note that your examples are largely overstating the problem. The prior is not typically as subjective as you are implying. Furthermore, the Bayesian prediction converges to the frequential result in the limit of 'large' data. If your results are so highly sensitive to your prior that the results drastically change, then you either have: 1) insufficient data, at which point the frequentist results would also certainly have been as bad or worse (the prior acts as a regularization) 2) a misspecified prior Either way, it is absolutely required that you perform tests to ensure your results are not greatly sensitive to your choice of prior, or if so that this is clearly noted as a modeling assumption. And bayes naturally has many means to test precisely this, such as cross-validation, bayes-factors, hierarchical, etc. I also note that this is why I typically prefer Jeffreys Priors in my work. I mention this so that one can see that not all priors are 'subjective'. These are difficult to use in some fields, admittedly.

by arcanus 11y ago

> such as it's extremely hard to calculate priors. I think you mean it is difficult to formulate priors? Typically, calculating a prior only involves sampling from a distribution.

Fomite 11y ago

Though technically, obtaining that distribution, if you're doing anything above "Eh, it's probably normal..." could be described as "hard".

eanzenberg 11y ago

Let's use this example: http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/ http://betterexplained.com/articles/an-intuitive-and-short-e... -1% of women have breast cancer (and therefore 99% do not). -80% of mammograms detect breast cancer when it is there (and therefore 20% miss it). -9.6% of mammograms detect breast cancer when it’s not there (and therefore 90.4% correctly return a negative result). For generating a prior, I posit it's really hard to determine that 1% of women have breast cancer, and this prior is almost linearly sensitive to whether you actually have cancer: prior | % likelihood of cancer 2% | 16% 1% | 7.8% 0.5% | 4% Why do I think it's hard to determine the prior? Imagine you are a 34 yrs old woman in SF. Do you use the rate of breast cancer for only 34 yr olds? For the range of 30-40 yr olds? 30-65 yr olds? For women only in SF? In the bay area? In California? Data from 2000-2016? 2010-2016? This becomes hard because positive cases are rare (thank god), so the population of which you calculate the prior from can be noisy. Imagine if the positive result occurs 1e-5 times.. you are talking about 10's of cases every million. Vary the prior probability and your posterior probability will vary just as much.

arcanus 11y ago
hmm, what you are discussing above is typically called the formulation of the prior, not the calculation of it. And I agree that this formulation can be problematic. It is almost certainly the most contentious element in Bayesian inference. However, I will note that your examples are largely overstating the problem. The prior is not typically as subjective as you are implying. Furthermore, the Bayesian prediction converges to the frequential result in the limit of 'large' data. If your results are so highly sensitive to your prior that the results drastically change, then you either have: 1) insufficient data, at which point the frequentist results would also certainly have been as bad or worse (the prior acts as a regularization) 2) a misspecified prior Either way, it is absolutely required that you perform tests to ensure your results are not greatly sensitive to your choice of prior, or if so that this is clearly noted as a modeling assumption. And bayes naturally has many means to test precisely this, such as cross-validation, bayes-factors, hierarchical, etc. I also note that this is why I typically prefer Jeffreys Priors in my work. I mention this so that one can see that not all priors are 'subjective'. These are difficult to use in some fields, admittedly.