what is the difference between Bayesian and non-Bayesian statistics?
I would summarize the difference as follows:
* 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 (A) given the null hypothesis B, 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 observed data (A), how confident are we that the null hypothesis (B) is true".
To simplify the logic- this is probability 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. It changes depending on the overall state of knowledge. At the same time, it is this probability 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 frequentist hypothesis test, no matter how many data-points 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 could 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.
Thus, classic statistic inference simply substitutes the main question of interest for another question, but assumes these questions are equivalent, P(B/A)=P(A/B). This is wrong.
The fundamental difference is between the frequentist and Bayesian understanding of a probability. For a frequentist, a probability is a hypothetically limiting relative frequency in a hypothetically infinite repetition of a change set up. As Neyman and Pearson pointed out, that means that we can never talk about the probability that any particular hypothesis is true. All that we can say probabistically is about the long run properties of a decision rule.
For a Bayesian, a probability is not some mythical limiting relative frequency, rather it is a measure of subjective belief. A Bayesian is therefore perfectly happy to talk about the probability that a particular hypothesis is true.
Now, both relative frequencies and subjective beliefs (constrained to require fair betting) must obey the same set of Kolmgorov's axioms. That means that both Bayesian and Frequentist statistics will often deliver the same conclusion.
But that is not always true. There are situations in which both approaches will
deliver very different results - read about group sequential clinical trials.
There are several differphilosophies of statistical inference, each with a different understanding of probability. They include frequentist, Fisherian, Structuralist and Likelihoodist inference.
Each philosophy has situations in which its Pplications just don't make sense. So most applied statisticians (like me) are a bit post moderntist. We swap between philosophies depending on the needs of the particular problem.
Hello. I have somewhere the book by Armitage and Berry; Statistical methods in medical Research. I will look for it, if you want and tell you the chapter to find. However the binding has deteriorated and the whole book is falling apart all the pages have fallen out. I didn't so much understand it anyway as it was a bit difficult to comprehend, some of it. Kind Regards.
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