Dear all,
I am interested in your assessment of these two approaches in empirical social research. Above all, I would be interested to know whether you assume that there will be a "paradigm shift" to Bayesian approaches and whether this will happen in the ongoing decade or not.
I can see the advantages of Bayesian statistics, but I am more or less reluctant to assume a quick fundamental change in the approaches.
What´s your expectation on this issue?
Greetings,
Michael
I think it has happened already. If you want to find estimates for a parametric model, it is much easier to obtain precise results using the Bayesian framework.
The posted question was the subject of extensive discussion in the literature. It comes down to the validity of the current prevalent statistical hypothesis testing methodology, and it has deep methodological implications. I would summarize the current state this way:
* currently prevalent frequentist approach is based on the single P value (or t-test, F-test, etc). It answers the question "How frequently / (probable) would the observed or stronger data (test statistics) occur if the trial repeated many times given the null hypothesis is true." To simplify and highlight the logic - this is the probability of the getting observed data given the null hypothesis, i.e. P(A/B).
* the real question that we want to know is different, it is the inverse probability, i.e. "Given the available data, how confident are we that the null hypothesis is true". To simplify the logic- this is probability (likelihood) of the null hypothesis given the observed data, i.e. P(B/A).
* In general, the conditional probability P(A/B) is NOT equal to the probability P(B/A).
* These inverse probabilities are very different in meaning; they are not based on frequency. This is known as subjective probability (likelihood). It changes depending on the overall state of knowledge. At the same time, it is this probability (likelihood) that is usually the primary interest in hypothesis testing.
* the 1-st approach is traditional frequentist one due to Fisher et al. The 2-nd approach is the Bayesian one.
* In frequentist approach the P value answers only a limited-value hypothetical question applied in the long run if the trial is repeated many times but not in the particular single trial. In contrast, Bayesian approach combines all the elements needed to answer the actual question of interest whether the null hypothesis true or not.
* it is unlikely that Bayesian approach will replace the prevalent frequentist one in the near future, mainly because of practical difficulties of deriving the prior probabilities before the test commencement.
* The good news that could serve as a consolation is that the frequentist and Bayesian approaches usually eventually converge to the same conclusion as experimental evidence accumulates for or against the hypothesis to be tested.
* The bad news is that it is usually unknown how long will it take for these two to eventually converge....
And even in convergence, for the Fisherian hypothesis test, no matter how many data you have, the p-value is the frequency you would expect to get that data set or worse with repeats of the experiment. With the Bayesian approach, you are measuring the uncertainty about the hypothesis being true. So, the numbers converge, but the interpretations still remain very different.
*It is measuring the uncertainty about the hypothesis being true is typically the primary goal of a research and/or data analysis. However, P-value (t-test) answers another hypothetical question that we only use as an unfortunate substitute for the actual question of interest.
"it is unlikely that Bayesian approach will replace the prevalent frequentist one in the near future, mainly because of practical difficulties of deriving the prior probabilities before the test commencement. "
> What if we use non-informative priors?
Since in many, if not most cases, the two approaches give the same answer, I would say no. However, as Andrey Davydenko remarks, as Bayesian Methods become easier to use, their use will increase. I don't know if that's a paradigm shift or just a desire to use the best available statistical methods for the current problem..
Best wishes, David Booth
more information in the link:
https://www.researchgate.net/post/When_should_we_apply_Frequentist_statistics_and_when_should_we_choose_Bayesian_statistics
(especially in the Daniel Wright's answer)
Roberto Molteni Your post was excellent. I was actually at Efron's talk but after 15 years had forgotten. I would agree with Efron. Best wishes, David Booth
I recently demonstrated that it was a mistake to use likelihood function in the Bayes Theorem, which leads to the so-called reformulated Bayes Theorem (by some authors) that is the basis of the Bayesian statistics. If this mistake is corrected, the frequentist and Bayesian inference may unify, and in fact, reunite. Certainly, this is an extraordinary claim. But please refer to the preprint that might have provided extraordinary evidence.
Huang H 2020 A new Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference, preprint, DOI: 10.13140/RG.2.2.35338.08646, available on ResearchGate: https://www.researchgate.net/publication/344552280_A_new_Bayesian_method_for_measurement_uncertainty_analysis_and_the_unification_of_frequentist_and_Bayesian_inference?channel=doi&linkId=5f7fd8a5458515b7cf71d5ec&showFulltext=true
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