I can use the phone locator on the base of the instrument to locate the phone and when I press the phone locator the phone starts beeping. I can hear the phone beeping. I also have a mental model which helps me identify the area from which the sound is coming. Therefore, upon hearing the beep, I infer the area of my home I must search to locate the phone. Now, apart from a mental model which helps me identify the area from which the sound is coming from, I also know the locations where I have misplaced the phone in the past.
So, I combine my inferences using the beeps and my prior information about the locations I have misplaced the phone in the past to identify an area I must search to locate the phone. A Bayesian defines a "probability" in exactly the same way that most non-statisticians do - namely an indication of the plausibility of a proposition or a situation. If you ask them a question about a particular proposition or situation, they will give you a direct answer assigning probabilities describing the plausibilities of the possible outcomes for the particular situation and state their prior assumptions.
A Frequentist is someone that believes probabilities represent long run frequencies with which events occur; if needs be, they will invent a fictitious population from which your particular situation could be considered a random sample so that they can meaningfully talk about long run frequencies. If you ask them a question about a particular situation, they will not give a direct answer, but instead make a statement about this possibly imaginary population. Many non-frequentist statisticians will be easily confused by the answer and interpret it as Bayesian probability about the particular situation.
However, it is important to note that most Frequentist methods have a Bayesian equivalent that in most circumstances will give essentially the same result, the difference is largely a matter of philosophy, and in practice it is a matter of "horses for courses". Frequentist: Sampling is infinite and decision rules can be sharp.
Data are a repeatable random sample - there is a frequency. Underlying parameters are fixed i. Bayesian: Unknown quantities are treated probabilistically and the state of the world can always be updated.
Data are observed from the realised sample. Parameters are unknown and described probabilistically. It is the data which are fixed.
There is a brilliant blog post which gives an indepth example of how a Bayesian and Frequentist would tackle the same problem. Why not answer the problem for yourself and then check? It ends up head 71 times. Then you have to decide on the following event: "In the next two tosses we will get two heads in a row. Let us say a man rolls a six sided die and it has outcomes 1, 2, 3, 4, 5, or 6.
Furthermore, he says that if it lands on a 3, he'll give you a free text book. The Frequentist would say that each outcome has an equal 1 in 6 chance of occurring. She views probability as being derived from long run frequency distributions.
The Bayesian however would say hang on a second, I know that man, he's David Blaine, a famous trickster! I have a feeling he's up to something. She views probability as degrees of belief in a proposition. The Bayesian is asked to make bets, which may include anything from which fly will crawl up a wall faster to which medicine will save most lives, or which prisoners should go to jail. He has a big box with a handle. He knows that if he puts absolutely everything he knows into the box, including his personal opinion, and turns the handle, it will make the best possible decision for him.
The frequentist is asked to write reports. He has a big black book of rules. If the situation he is asked to make a report on is covered by his rulebook, he can follow the rules and write a report so carefully worded that it is wrong, at worst, one time in or one time in 20, or one time in whatever the specification for his report says. The frequentist knows because he has written reports on it that the Bayesian sometimes makes bets that, in the worst case, when his personal opinion is wrong, could turn out badly.
The frequentist also knows for the same reason that if he bets against the Bayesian every time he differs from him, then, over the long run, he will lose.
In plain english, I would say that Bayesian and Frequentist reasoning are distinguished by two different ways of answering the question:. Most differences will essentially boil down to how each answers this question, for it basically defines the domain of valid applications of the theory. Now you can't really give either answer in terms of "plain english", without further generating more questions.
For me the answer is as you could probably guess. Additionally, the calculus of probabilities can be derived from the calculus of propositions. This conforms with the "bayesian" reasoning most closely - although it also extends the bayesian reasoning in applications by providing principles to assign probabilities, in addition to principles to manipulate them.
Of course, this leads to the follow up question "what is logic? Logic has all the same features that Bayesian reasoning has. For example, logic does not tell you what to assume or what is "absolutely true". It only tells you how the truth of one proposition is related to the truth of another one. You always have to supply a logical system with "axioms" for it to get started on the conclusions. They also has the same limitations in that you can get arbitrary results from contradictory axioms.
For me, to reject Bayesian reasoning is to reject logic. For if you accept logic, then because Bayesian reasoning "logically flows from logic" how's that for plain english :P , you must also accept Bayesian reasoning.
I wanted to add into the frequentist answer that the probability of an event is thought to be a real, measurable observable? But I couldn't do this in a "plain english" way. So perhaps a "plain english" version of one the difference could be that frequentist reasoning is an attempt at reasoning from "absolute" probabilities, whereas bayesian reasoning is an attempt at reasoning from "relative" probabilities.
Another difference is that frequentist foundations are more vague in how you translate the real world problem into the abstract mathematics of the theory. A good example is the use of "random variables" in the theory - they have a precise definition in the abstract world of mathematics, but there is no unambiguous procedure one can use to decide if some observed quantity is or isn't a "random variable".
The bayesian way of reasoning, the notion of a "random variable" is not necessary. A probability distribution is assigned to a quantity because it is unknown - which means that it cannot be deduced logically from the information we have.
This provides at once a simple connection between the observable quantity and the theory - as "being unknown" is unambiguous. You can also see in the above example a further difference in these two ways of thinking - "random" vs "unknown". Conversely, "being unknown" depends on which person you are asking about that quantity - hence it is a property of the statistician doing the analysis.
This gives rise to the "objective" versus "subjective" adjectives often attached to each theory. It is easy to show that "randomness" cannot be a property of some standard examples, by simply asking two frequentists who are given different information about the same quantity to decide if its "random". One is the usual Bernoulli Urn: frequentist 1 is blindfolded while drawing, whereas frequentist 2 is standing over the urn, watching frequentist 1 draw the balls from the urn.
If the declaration of "randomness" is a property of the balls in the urn, then it cannot depend on the different knowledge of frequentist 1 and 2 - and hence the two frequentist should give the same declaration of "random" or "not random". In reality, I think much of the philosophy surrounding the issue is just grandstanding. That's not to dismiss the debate, but it is a word of caution. Sometimes, practical matters take priority - I'll give an example below.
A senior colleague recently reminded me that "many people in common language talk about frequentist and Bayesian. I think a more valid distinction is likelihood-based and frequentist.
Both maximum likelihood and Bayesian methods adhere to the likelihood principle whereas frequentist methods don't. We have a patient. The patient is either healthy H or sick S. If the patient is sick, they will always get a Positive result. So far so good.
Even if all you have in your hand is a single CI you just calculated and you specifically want to frame the probability in terms of that CI, then you would do something like:. You see how in no way does it follow that P A B should equal 0. I myself have had my struggles too. Hey, I think I got it now! You are right about that Matrix analogy! It is primarily about the data. However, that seems to be a backward and counter-intuitive type of investigation. Well, among many other things, it does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does!
Like I said in an earlier reply, the important thing really is to remember the kind of philosophical and mathematical framework frequentist statistics is based in, in order to not make faulty conclusions. Don't get too hung up on any one particular hypothesis - if you're wrong, you will find out sooner or later. Anyway, thanks for all the questions.
I think this was a very good discussion that is going to be helpful to anybody who is having similar difficulties with interpreting results of frequentist analyses. Hi, thank you for the article! Best, Ola. The maximum likelihood estimate of a parameter is not the same as the true value of the parameter. Then say you take a sample from that population and calculate its mean which happens to be The maximum likelihood estimate is basically like a guess for the true mean.
There are other types of estimates of population parameters from sample data like the maximum a posteriori estimate , which are more than just the mean of the sample. I will talk about those in future posts. You raise an interesting question! Is the mean really more likely to be near the center of a confidence interval than near the edges? This means that the whole question about the probability of the mean being closer to the center or the edges of a CI is not defined under this framework.
To make any probabilistic claims about the value of the mean, you need to be operating under the Bayesian framework. Can we then say that there is a higher probability that the mean is closer to the center than to the edges of a CI?
Take another look at the section A graphical illustration of calculating confidence intervals in the main text. In the animated simulation, pay attention to the confidence intervals that cover the true mean. In order for the original claim to be true, it should generally be the case that of the CIs that cover the true mean, a higher percentage will be of the CI-center type than of the CI-edge type. But the procedures followed for constructing a confidence interval make no such promise!
Even though conventions do exist, there are many procedures that would construct a confidence interval and different procedures will generally produce different confidence intervals for the same sample data. Some of them might indeed lead to a higher CI-center type CIs than others, but this is by no means a requirement that a CI generating procedure must follow. Thanks for your response.
What about the normal approximation to the binomial distribution? Is that confidence interval more likely near the center? Whether this is the case will depend on the parameter p of the binomial distribution, as well as on the sample size and the desired confidence level.
And, the wider the CI is, the less likely it is that the true mean will be near the edges. In other words, because the sample mean is an unbiased estimator of the true population mean, you can generally expect that the population mean will be closer to the sample mean and hence, to the center of the CI.
But you can easily have a case where the true mean happens to be closer to the edges than to the center of a confidence interval. For example, if the population mean the parameter p is around 0. It will all depend on the factors I mentioned above sometimes in complicated ways. Bottom line is that confidence intervals were never intended to provide information about the question we started with. Once you have a CI, you have no reason to think that any value inside the interval is a better candidate to be the true mean compared to the others.
Do these explanations help you in understanding my first answer to your original question? Let me know if you need clarification about any part of them.
Thanks for raising this question! Hi, I think I am late but I just I want to encourage you to publish that follow-up entry on why the mathematics are different in the two approaches and the truth is not anywhere in the middle. Ah, yes. That post you mention is on my very long list of drafts. I will surely get to it at some point!
Hello, thanks for the really insightful posts! Could you maybe give a more concrete example of data prediction from both the frequentist and bayesian perspectives? At the end of your post, when you talk about p-values and confidence intervals, it seems this focused just on parameter estimation, or did I misinterpret this?
Is frequentist NHST considered model comparison could the null and alternative hypotheses be thought of as two different models , and if this not always true, could you explain why and provide an example of a frequentist approach to model comparison? Hello, Jai! But let me try to clarify things for you here. Say you have the heights of randomly picked adult females. So let me fit a normal distribution to the the data above. Fitting a normal distribution basically means to find the mean and standard deviation that would best fit the observed data.
You do MLE and determine that the best fitting normal distribution is one with a mean of cm and a standard deviation of 8 cm. What do you think is the probability that she will be tall when she grows up? Well, I consider a height of cm or more to be tall for a woman. Therefore, the probability of your daughter reaching a height of cm or more is the integral from to positive infinity over the normal distribution with those parameters.
Which is around 0. Your initial assumption that female height has a normal distribution is probably wrong. I think that height actually has a gamma distribution. Your parameter estimation is irrelevant because you should have done it with a gamma distribution instead, not a normal distribution. So, your prediction is probably wrong too. Well, the process of resolving this disagreement would be an example of model comparison. You see, when you do parameter estimation, you do it in the context of some more basic assumptions.
In this case, the assumption was that of the normal distribution as opposed to other distributions. The set of all these basic assumptions is your model. An alternative model would be anything that has one or more different assumptions.
In the example above, the different assumption is that the distribution is a gamma distribution with parameters shape and scale , not a normal distribution with parameters mean and variance.
Model comparison essentially means determining which model fits the data better. That is, if you take the best possible fit of each model, which one seems like a more adequate fit? But let me try to give you some basic intuition.
In parameter estimation, you consider how good of a candidate each possible value of the parameter is to fit the data, right? Bayesians would construct a probability distribution over the parameter space, whereas frequentists would calculate a point estimate MLE. Now, in model comparison, Bayesians would instead build a probability distribution over the model space. If there are only 2 candidate models, you would calculate the probability of each model, given the data, in a Bayesian way by having some prior distribution over the models.
Then there are different ways to select the best model based on those probabilities. On the other hand, frequentists will not bother trying to assign probabilities to the different models because in this framework probabilities apply only to data, not to things like parameters or models.
One naive approach to frequentist model selection would be to directly compare the MLEs of the two models. Whichever model gives the highest maximum likelihood, it is to be selected.
But with that approach you would quickly hit one of the most serious problems in the field: that of overfitting. There are more sophisticated methods that frequentists would use to do model comparison, like cross-validation and likelihood-ratio test. And yes, you can view NHST as a type of model comparison. In a typical frequentist setting, you would have one of your models the simpler one be the null hypothesis and the other model be the alternative hypothesis.
I hope I managed to answer your question. Feel free to subscribe to my mail list if you want to get a notification when that happens. Terrific, this post and comment has made things much clearer for me! Thanks again!! Looking forward to the future posts :. I just stumbled upon your website and I love it.
The topics are interesting, and the explanations are lucid. I would argue that there is nothing inconsistent about these results. The reason is that the true mean is unknown, and unless you do a census of the population, unknowable.
Based on each sample therefore, a different underlying population is inferred, and without a census of the population, you have no basis to prefer one inferred population to another.
Thus, the following statement seems to me not valid. But, as suggested by my argument below, even that would only imply that that the population which one would infer from second sample has a smaller variance than the population which one would infer from first sample.
Thus, neither [, ] nor [, ] is a proper subset of [, ]. They are different, independent samples. The population implied by the latter sample is different than the population implied by the former sample.
Since the true population is unknown, there is no basis for saying that one sample is more correct than the other or that one is a subset of the other sample.
Each of the samples implies a certain underlying population; in particular, each sample implies a population which has a different mean and variance than that implied by the other samples, but since the true parameters are unknown and unknowable, there is nothing inconsistent about different samples implying different underlying populations.
I have long had an interest in these topics, although it has been dormant for some years. I look forward to a response from you in the hope that I can learn more about these things.
Hi David, thank you for your nice words about the website! Hopefully you will find it even more useful and interesting as my news posts arrive! Yes, good catch! I definitely meant there, not I edited my old comments to fix the error. But they are all samples of the same population. There are no different populations here.
If your point is that in my particular hypothetical example we could potentially be dealing with different populations, just imagine an even stricter scenario. One of the confidence intervals is [, ] and the other is [, ]. If, based on these two confidence intervals, we try to make the following statements about the underlying population P:.
Hypothetically, the true mean could be, say, cm, and it would be possible to draw samples of out of 50, to give the results you posited. I am trying to relate this difference in interpretation to the larger point of your original post. In so doing, I decided that I do not fully understand this paragraph:. You wrote: If your point is that in my particular hypothetical example we could potentially be dealing with different populations.
I think that is my point. So long as you have not done a census of the population, the true population, and thus the true mean, must be unknown and unknowable.
I will posit a wild hypothetical which might lead to a clarification. Suppose out of a population of 50, you take a random sample of 49,, and based on that sample you calculate an estimate of the mean lying between The obvious answer is that the last item in the population, which by chance we did not include in our sample, had a value in excess of , Now, as mere numbers in a population, this would be entirely possible. But if you say the population was human beings and the values were heights, then you can say that the result is absurd.
You would throw out the last value as an outlier. In this case, you are using additional information beyond the sample taken. When you say that the population consists of heights of human beings, you are adding additional information about the range of the population, to wit, that a value of , cm is outside of the allowable range, and must not be taken into account.
This additional information may be the kind of information which justifies an assumed a priori probability in a Bayesian approach. If you draw many samples out of the same population, some of them will have very different means.
All in all, one method is not better than the other; what matters is understanding the underlying logic of each or seeking advice from someone who is familiar with both. Overview Bayesian versus frequentist statistics: what are the differences? Bayesian approach: benefits and limitations Frequentist approach: benefits and limitations Which approach should you choose, frequentist or Bayesian?
Frequentist statistics , which could also be described as experimental or inductive, relies on the law of observations. Essentially, if the coin is balanced, then in theory the probability of previous experiments. Topics covered by this article. He is a recognized expert in AI innovation and software development.
In his posts he shares his vision of the market and his technology expertise. Recommended articles for you. You may also enjoy these articles. Joseph Solomon - November 8, Conversion Rate Optimization.
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