Frequentists can use confidence intervals to partially solve this problem. When the number of flips is higher, the range of values covered by the interval is going to be much tighter. In other words, if your confidence interval is [0. When the number of flips is low, this can make a big difference. As I said, this is a tricky topic and I am probably not covering everything to fully answer your question.
Please feel free to ask for further clarification! I enjoyed this post very much. To be honest, my knowledge of statistics is limited to basic classes in college and some in grad school, and I still am learning. I never understood what the MLE was, and your explanations are clear and concise without omitting crucial details. The overall distinction between Bayesian and Frequentist approaches was clear to me as well.
Thank you for your observation. Thank you for this post. Hi Joseph, very good question. I know what I wrote sounds counter-intuitive.
It used to be counter-intuitive for me for a long time. But stay with me and I promise, once it clicks, you will have a small feeling of just been taken outside of the Matrix. The first problem is something I already pointed to earlier in the post. Granted, this is a philosophical, not mathematical problem, but this view is at the root of the Frequentist approach and was passionately defended by founders like Fisher, Pearson, and Neyman. The answer is still no, and this time the problem is also mathematical.
Say you have a sample of 20 students out of students in a particular school and you measure their height. Then you draw another random sample of 20 students from the same school and calculate a confidence interval of [, ].
Then you draw a third sample and this time you calculate a confidence interval of [, ] just happened to pick really short people. So what? The post is informative and has piqued my interest to understand Bayesian analysis. I wanted to know when we report effect sizes based and confidence intervals on full model-averaged parameter estimates, does it fall in the frequentist-based approach? Thank you, Harshad. Confidence intervals do come from the domain of frequentist statistics.
However, effect sizes themselves are sort of framework agnostic when it comes to the Bayesian vs. By that I mean that you can certainly use them in both frameworks, but in a different manner. They are simply unitless measures of the size of a particular difference. Once you have them, you can treat effect sizes themselves as random variables and do a Bayesian estimate of them. Or you can construct confidence intervals around them and then be in the domain of frequentist statistics.
Does my answer make sense? And, more importantly, did I understand your question correctly? If not, please provide a bit more details. Sure, Salah. The parameters I was talking about in the main text are exactly distribution parameters. Wonderful blog! I hope you still keep an eye on the replies. Now my intuitive response would be that this is not meaningful.
I imagine having a jar of five red marbles and ninety-five black ones, a hundred in total. Blindfolded, I stick my hand in the jar and grab a random marble. Now, without opening my fist or taking off my blindfold, I guess what chance I have of having a black marble the true mean inside my closed fist the confidence limits.
I already answered a similar question. But let me try to expand on this issue a bit more here. The process of collecting sample data and calculating the mean and a confidence interval around it guarantees that: if you repeat this process and the whole process!
This is all the confidence interval promises. It makes no claims regarding the probability of any specific confidence interval covering the mean. Remember, the actual mean is some fixed but unknown value and not a random variable!
In this context, probability is simply used to measure our uncertainty about where the actual mean is. Each of them is an equally good candidate. Each Xk stands for a real number like How would you go about calculating this probability in the general case? Which, probabilistically speaking, translates to you having a prior distribution over the possible values of the mean of X.
What you would do in this situation is basically calculate the definite integral over the said normal distribution, where the lower bound of the integral is 15 and the upper bound is Are you with me so far?
Now that we have new data, we want to calculate the posterior probability:. But, think about it. It calculates exactly the probability that the mean is between those two values, given the data we have the evidence.
Well, for one, notice that the prior distribution you start with is a strong factor in this calculation. Thank you for replying, great! Sorry for mssing the earlier question that looked like the one I had. I feel new questions forming already, though. I accept that you are right, without me really seeing why yet. But this leads me to my main concern: What is then the use of CIs as a tool to understanding your data?
It is abstract, mathematical, counter-intuitive etc. How does it help anyone make a decision? One scientist told me with a straight face that a p-value from a single study is meaningless.
Well, all studies are single studies. All of this is important for inference, evidence, decision making which Is my real point of interest. Consider a jar with only two marbles, one red and one black. You grab one marble with each hand.
What is the chance that the marble is in your left hand? It is 0. In your mathematical example you show that calculating the probability of finding the true mean has different results when you take the data distribution into account compared with not using the distribution. Well, here you have basically touched upon some common points of criticism towards frequentist statistics.
Indeed, frequentists historically strongly resisted the intuitive temptation to assign probabilities to hypotheses.
And to any non-repeatable event in general. The reason that scientist told you that a p-value from a single study is meaningless is probably because he was implicitly making the same argument.
Its only role is to be a sort of criterion for whether to reject or not reject your null hypothesis. Is this good enough? Well, empirically this has been a matter of different taste and historically, a matter of heated debates!
And I would personally agree with you, as I am also interested in those kinds of questions. Okay, let me try to adapt your marble example to confidence intervals.
Every time a new person between ages 20 and 30 enters the mall, you measure their height. Then you write down the boundaries of the interval, put the paper inside the marble, and throw the marble in a jar.
Then you reset your counts and the next person to enter becomes the first in your next sample and you repeat this process many times and have a new marble for each sample of consecutive people. Say you play this game for a full year and manage to collect 1 million marbles, each containing a confidence interval. You open it and you see that it holds the confidence interval [, ].
Think about it. For example, maybe there was a basketball tournament in the city that day and the players of all teams decided to go to the mall together after the tournament.
The first one is about a repeatable process. Namely, randomly picking a marble, reading its CI, and checking if it contains the true mean. The second one is about a hypothesis. Namely, the hypothesis that the true mean is between and Thank you so much for taking the time to replying at such length! Also grateful for the personal notification that you had posted a reply for me. To the first part, you are right that I feel something is missing to the frequentist picture of things.
His views seem waterproof, but somehow unproductive. But my in my mind the framework of the thought experiment was that our data analysis yields fists, not marbles. If one was forced to gamble onthe CI, taking the bet at odds of 95 to 5 and up is then a win in the long run. The Monty Hall problem analogy here would be to open one hand and see if the red marble is there, and then change your mind about the probabilities concerning the other fist. Notice that there is no mention of any specific values for the boundaries of the CI.
But if you have actual values for the boundary of the interval, then whether the mean is within that interval or not becomes a different independent question, unrelated to the CI you just calculated.
The interval I got was [, ]. Well I just did the same thing and I got an interval [, ]. Actually, just an hour ago I did the same thing you did and got a confidence interval of [, ]. Now, I ask you.
Are all those statements mutually consistent? If you say yes, then you would be violating the axioms of probability theory. Nor are they intended to. But even if we ignore that philosophical issue and construct a probability distribution for the true mean, then you would do so by collecting the same data as the one you collected for constructing the confidence interval in the first place.
And, at each moment, you can calculate the probability of the true mean being within a certain interval by integrating the posterior over that interval. Then you can make actual probabilistic statements about the true mean being within a certain interval.
What is the probability that the true mean is between X and Y? But if you want to make probabilistic statements about a specific interval, then you need more. 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 Rationalizing this distribution to the extreme to make it a simple [ The cut-off on the [ This approach is based solely on the data from tests run in strictly similar conditions for each variation hence its reputation as a data-driven method.
One of the most rigorous analyses comparing the frequentist and Bayesian approaches was carried out by the statistician Valen Johnson and summarized in his article published in the Proceedings of the National Academy of Sciences in 1. The aim of his frequentist analysis was to explore the data collected so as to identify a significant effect that could only be explained by the hypothesis of the experiment.
His Bayesian analysis compared two hypotheses and assessed the chances that one was true in comparison with the other, by using the data available at the time of the experiment and the information already known about the subject. No, because the Bayesian method has significant advantages when circumstances allow. Generally speaking, this question of which method is better, the Bayesian or the frequentist, is subject to ongoing debate amongst experts and extends far beyond the immediate needs of marketing teams.
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. 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. Those are the statements that would be make by a frequentist.
Those statements are quite simple to understand and are true. There's no need to waffle about a 'frequentist interpretation'. But, things get interesting when you try to turn things around. Given the test result, what can you learn about the health of the patient? Given a negative test result, the patient is obviously healthy, as there are no false negatives. But we must also consider the case where the test is positive. Was the test positive because the patient was actually sick, or was it a false positive?
This is where the frequentist and Bayesian diverge. Everybody will agree that this cannot be answered at the moment. The frequentist will refuse to answer.
The Bayesian will be prepared to give you an answer, but you'll have to give the Bayesian a prior first - i. If you are satisfied with statements such as that, then you are using frequentist interpretations. This might change from project to project, depending on what sort of problems you're looking at. This requires a prior and a Bayesian approach. Note also that this is the only question of interest to the doctor. The doctor will say "I know that the patients will either get a positive result or a negative result.
I also now that the negative result means the patient is healthy and can be send home. The only patients that interest me now are those that got a positive result -- are they sick?. To summarize: In examples such as this, the Bayesian will agree with everything said by the frequentist. But the Bayesian will argue that the frequentist's statements, while true, are not very useful; and will argue that the useful questions can only be answered with a prior.
A frequentist will consider each possible value of the parameter H or S in turn and ask "if the parameter is equal to this value, what is the probability of my test being correct? Bayesian and frequentist statistics are compatible in that they can be understood as two limiting cases of assessing the probability of future events based on past events and an assumed model, if one admits that in the limit of a very large number of observations, no uncertainty about the system remains, and that in this sense a very large number of observations is equal to knowing the parameters of the model.
Assume we have made some observations, e. In Bayesian statistics, you start from what you have observed and then you assess the probability of future observations or model parameters. In frequentist statistics, you start from an idea hypothesis of what is true by assuming scenarios of a large number of observations that have been made, e.
It is only then that you take your actual outcome, compare it to the frequency of possible outcomes, and decide whether the outcome belongs to those that are expected to occur with high frequency. Otherwise, you conclude that the observation made is incompatible with your scenarios, and you reject the hypothesis.
Thus Bayesian statistics starts from what has been observed and assesses possible future outcomes. Frequentist statistics starts with an abstract experiment of what would be observed if one assumes something, and only then compares the outcomes of the abstract experiment with what was actually observed.
Otherwise the two approaches are compatible. They both assess the probability of future observations based on some observations made or hypothesized. Positioning Bayesian inference as a particular application of frequentist inference and vice versa.
I would say that they look at probability in different ways. The Bayesian is subjective and uses a priori beliefs to define a prior probability distribution on the possible values of the unknown parameters.
So he relies on a theory of probability like deFinetti's. The frequentist see probability as something that has to do with a limiting frequency based on an observed proportion. This is in line with the theory of probability as developed by Kolmogorov and von Mises. A frequentist does parametric inference using just the likelihood function. A Bayesian takes that and multiplies to by a prior and normalizes it to get the posterior distribution that he uses for inference.
The simplest and clearest explanation I've seen, from Larry Wasserman's notes on Statistical Machine Learning with disclaimer: "at the risk of oversimplifying" :. What's tricky is that we work with two different interpretations of probability which can get philosophical. There is nothing subjective about this which can be viewed as a good thing, however we can't really perform infinite flips and in some cases we can't repeat the experiment at all, so an argument about limits or long run frequencies might be in some ways unsatisfactory.
On the other hand, the Bayesian viewpoint is subjective, in that we view probability as some kind of "degree of belief", or "gambling odds" if we specifically use de Finetti's interpretation. For example, two people may come into the coin flipping experiment with different beliefs about what they believe about the coin prior probability.
In practice, statisticians can use either kind of methods as long as they are careful with their assumptions and conclusions. Nowadays Bayesian methods are becoming increasingly popular with better computers and algorithms like MCMC. Also, in finite dimensional models, Bayesian inference may have same guarantees of consistency and rate of convergence as frequentist models. I don't think there is any way around really understanding Bayesian and frequentist reasoning without confronting or at least acknowledging the interpretations of probability.
The way I answer this question is that frequentists compare the data they see to what they expected. That is, they have a mental model on how frequent something should happen, and then see data and how often it did happen. Bayesian people, on the other hand, combine their mental models.
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