The first attempt at mathematical rigour in the field of probability, championed by Pierre-Simon Laplace, is now known as the classical definition.Developed from studies of games of chance (such as rolling dice) it states that probability is shared equally between all the possible outcomes, provided â¦ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. Dichotomous thinking crept in. Frequentists only allow probability statements about sampling. Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials. Those approaches are: This approach traces back to the field where probability was first sistematically employed, which is gambling (flipping coins, tossing dice and so forth). 1 Learning Goals. We have now learned about two schools of statistical inference: Bayesian and frequentist. I think the question Bayesian *versus* frequentist is wrong. In fact Bayesian procedures often have good frequentist properties. However, prerequisites are essential in order to appreciate the course. Calculus for Data Science and ML: Integrals, Finding the Equation of a Line Tangent to a Function, The assumption of symmetry is far too strong and irrealistic. But as you can see, it can run into some deep philosophical issues. Be able to explain the diï¬erence between the p-value and a posterior probability to a doctor. Bayesian vs frequentist: estimating coin flip probability with frequentist statistics. It is surprising to most people that there could be anything remotely controversial about statistical analysis. Kudos to Roy for coming up with example, and shame on me for screwing up the initial posting! All these rolls are not going to change that. There are three different frameworks under which we can define probabilities. (A less subjective formulation of Bayesian philosophy â¦ It isnât science unless itâs supported by data and results at an adequate alpha level. Since it is impossible, the probability is equal to zero and not 1/6. We could ask other questions, for example, is this a fair die? Class 20, 18.05 Jeremy Orloï¬ and Jonathan Bloom. Sometimes we also get interpretations that are not particularly intuitive. Depending upon what we know about the universe, we might get different answers. A fantastic example taken from Keith Winstein's answer found here: What's the difference between a confidence interval and a credible interval? A fantastic example taken from Keith Winstein's answer found here: What's the difference between a confidence interval and a credible interval? To view this video please enable JavaScript, and consider upgrading to a web browser that It can be read as the probability of A, given that B is the case. One of these is an imposter and isnât valid. The frequentist view defines probability of some event in terms of the relative frequency with which the event tends to occur.. Bayesian versus Classical (frequentist) Statistics. Basically, what in other approaches was a rule, in the subjective approach is an option. In Lesson 2, we review the rules of conditional probability and introduce Bayes’ theorem. We might be comparing routers from two different companies. 3. Let’s think about the previous example of the dice. The lectures provide some of the basic mathematical development as well as explanations of philosophy and interpretation. 16 answers. In that case, we can consider this infinite collection and ask what fraction of this infinite collection have universes that expand forever? We can ask more existential questions such as, what's the probability that the universe goes on expanding forever? â¢ Conceptually simple ... many outcomes. Frequentist probability and frequentist statistics. And so either it is fair, or it isn't fair. You can't pass this course unless you have understood the material. You may assume that I'm familiar with the material in Casella and Berger. Can someone give a good rundown of the differences between the Bayesian and the frequentist approach to probability? And the case of a specific fixed sample, when the data do not change, we will either always capture the true parameter or never capture it. If you indicate that price as π(E, S), the probability of event E is given by: Imagine you want to predict the probability that your favorite football team will win the match tomorrow. This interpretation supports the statistical needs of many experimental scientists and pollsters. For example, the probability of rolling a dice (having 1 to 6 number) and getting a number 3 can be said to be Frequentist probability. The second, there's a Frequentist framework, and the third one is a Bayesian framework. Say you wanted to find the average height difference between all adult men and women in the world. This course introduces the Bayesian approach to statistics, starting with the concept of probability and moving to the analysis of data. Sometimes the objectivity is just illusory. Frequentist statistics only treats random events probabilistically and doesnât quantify the uncertainty in fixed but unknown values (such as the uncertainty in the true values â¦ The purpose of this post is to synthesize the philosophical and pragmatic aspects of the frequentist and Bayesian approaches, so that scientists like myself might be better prepared to understand the types of data analysis people do. This means you're free to copy and share these comics (but not to sell them). 73 Citations. ... $\begingroup$ Very often in text-books the comparison of Bayesian vs. That would be an extreme form of this argument, but it is far from unheard of. Finally, inputting all values into the equation, we get a posterior probability for H 0 â 0.98. One of the first things a scientist hears about statistics is that there is are two different approaches: frequentism and Bayesianism. Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in a large number of trials. Read/Download File Report Abuse. The classical approach is pretty intuitive, nevertheless it suffers from some pitfalls: This approach was formally introduced in the field of natural science, where the assumption of symmetric position poorly fails. That's very difficult to apply in any of these other cases. For example, what is the probability that it rains tomorrow? The Bayesian approach allows direct probability statements about the parameters. Bayesian Statistics: From Concept to Data Analysis, Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. A real statistician (frequentist or Bayesian) would probably demand a lower p-value before concluding that a test shows the Sun has exploded; physicists tend to use 5 sigma, or about 1 in 3.5 million, as the standard before declaring major results, like discovering new particles. Those who criticize Bayes for having to choose a prior must remember that the frequentist approach leads to different p-values on the same data depending on how intentions are handled (e.g., observing 6 heads out of 10 tosses vs. having to toss 10 times to observe 6 heads; accounting for earlier â¦ This Classical approach works really well and we have equally likely outcomes or well-defined equally likely outcomes. This course combines lecture videos, computer demonstrations, readings, exercises, and discussion boards to create an active learning experience. Hence, probability does depend on the available information (the intuition will be clearer in the subjective approach), Again, there is one big assumption which is the convergence property of the frequency, whose limit might not exist, Repeating experiments under equivalent conditions might not be possible, There are events extremely rare, for which is impossible to run many simulations (think about extreme natural events like. In order to illustrate what the two approaches mean, letâs begin with the main definitions of probability. Which is the price you would be willing to pay to participate? For Alice, the answer is simple: the probability is 100% if the penny is in her left hand and 0% if itâs in her right hand. To view this video please enable JavaScript, and consider upgrading to a web browser that, Lesson 1.1 Classical and frequentist probability, Lesson 1.2 Bayesian probability and coherence. I think some of it may be due to the mistaken idea that probability is synonymous with randomness. And so frequentists are concerned with the probability of seeing a particular data sample given the null hypothesis and that's what the P value gives you. Frequentist Bayesian Estimation I have 95% confidence that the population mean is between 12.7 and 14.5 mcg/liter. Similarly, the event “five or six or one” (that is, the event in which one of those numbers turns out) represents 3 outcomes out of 6, hence the probability will be 3/6=0.5. âThe difference between frequentist and Bayesian approaches has its roots in the different ways the two define the concept of probability. This approach works great when we can define a hypothetical infinite sequence. As you can see, we obtained two different probabilities (0.5 vs o.55) for the same event. The essential difference between Bayesian and Frequentist statisticians is in how probability is used. while frequentist p-values, confidence intervals, etc. Now you decide to follow the empirical approach, and you start tossing your coin several times, let’s say 100. Imagine a lottery where you can win an amount of money equal to S if event E occurs. Great course. The probability of occurrence of an event, when calculated as a function of the frequency of the occurrence of the event of that type, is called as Frequentist Probability. One is the gracious invitation of Professor Jaakko Hintikka to contribute to the issue of his journal especially given to foundations of probability and statistics. In Lesson 1, we introduce the different paradigms or definitions of probability and discuss why probability provides a coherent framework for dealing with uncertainty. The idea of the classical approach is that, given a collection of k elements out of n (where 0≤k≤n), the probability of occurrence of the event E represented by that collection is equal to: To give you the intuition, let’s imagine you are tossing a dice and you want to predict the probability of the following collection of outcomes: We know that the n possible outcomes are 6. Frequentists use probability only to model certain processes broadly described as "sampling." ... Bayesian vs Classical Statistics? Well, if we have a particular physical dye, and we're asking, is it a fair die, then we can roll it a lot of times, but that's not going to change whether or not it's a fair die. Under the Classical framework, outcomes that are equally likely have equal probabilities. I didnât think so. In particular, the Bayesian approach allows for better accounting of uncertainty, results that have more intuitive and interpretable meaning, and more explicit statements of assumptions. The idea of the clasâ¦ So there are a total of 3 possible outcomes out of 36 equally likely outcomes, and so that's a probability of 1 in 12. On a side note, we discussed discriminative and generative models â¦ There is a 95% probability that the population mean is in the interval 136.2 g to 139.6 g. Hypothesis Testing If H0 is true, we would get a result as extreme as the data we saw only 3.2% of the time. Ask Question Asked 6 years ago. © 2020 Coursera Inc. All rights reserved. This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. Or, in the case of asking is this a fair dye? 1. The intuitive answer is 50%, as he has no knowledge about what hand the penny could be in. One of the ways to deal with uncertainty, in a more quantified way, is to think about probabilities. More details.. The p-value is highly significant. Be able to explain the diï¬erence between the p-value and a posterior probability to a doctor. But recently, so-called best-system interpretations of chance have become increasingly popular and important. You have the possibility to participate in a lottery where, if the team wins, you obtain a prize of 1000€, otherwise you gain nothing. The MDL, Bayesian and Frequentist schools of thought differ in their interpretation of how the concept of probability relates to the real world.. Probability. Imagine you want to know the probability of the outcome of your tossed coin being “head”. (Update based on Foster's comment below: instead of using the uniform distribution as a prior, we can be even more agnostic. INTRODUCTION The present paper is prompted by two stimuli. And how do we make the decisions in the presence of it? 2 Introduction. Well if we think about this, how many equally likely outcomes are possible on a pair of dice? Would you measure the individual heights of 4.3 billion people? The bread and butter of science is statistical testing. A very good introduction to Bayesian Statistics.Couple of optional R modules of data analysis could have been introduced . But then we can ask other questions, and they become more complicated under this approach. Frequentist Bayesian Estimation I have 95% confidence that the population mean is between 12.7 and 14.5 mcg/liter. This article will help you to familiarize yourself with the concepts and mathematics that make up inference. Frequentist vs Bayesian statistics- this has been an age-old debate, seemingly without an end in sight. Hence, given n random experiments run under equivalent conditions, we define the frequency of “success” (which is an event E) as: If we consider the “Empirical Law of Change”, which states that the more n increases, the more stable the frequency becomes, we can conclude that the limit of that frequency, for n->infinite, does exist and it is equal to the probability of the event “success”: Let’s size the difference between the frequency-based and classical approach with the following example. The Quizzes are also set at a good level. Under the Classical framework, outcomes that are equally likely have equal probabilities. 1. This article on frequentist vs Bayesian inference refutes five arguments commonly used to argue for the superiority of Bayesian statistical methods over frequentist ones. To participate, you have to buy one ticket. We will learn about the philosophy of the Bayesian approach as well as how to implement it for common types of data. These two approaches or philosophies are the two arms of inferential statistics, the branch of statistics that allows generalizations to be made about entire populations of data based on observations of some amount of sample data. For some reason the whole difference between frequentist and Bayesian probability seems far more contentious than it should be, in my opinion. Steven de Rooij, Peter D. Grünwald, in Philosophy of Statistics, 2011. This is a preview of subscription content, log in to check access. Question. As per this definition, the probability of a coin toss resulting in heads is 0.5 because rolling the die many times over a long period results roughly in those â¦ Probability can be defined as a tool to manage uncertainty. The possible outcomes of this scenario are two: having a car accident or not having a car accident. Given that, In this approach, there is no space for the concept of information, which is strictly related to probability. Oh, no. We could interpret it as a classical long run frequentist probability, but this means interpreting it like a confidence interval. the quotient P(B|A)/P(B) represents the support B provides for A. The Bayesian view defines probability in more subjective terms â as a measure of the strength of your belief regarding the true situation. Let’s say you are very confident about your team capabilities and you are willing to pay 700€. Your first idea is to simply measure it directly. This video provides an intuitive explanation of the difference between Bayesian and classical frequentist statistics. Statistics, Bayesian Statistics, Bayesian Inference, R Programming. The frequentist definition of probability allows to define a probability for the confidence interval procedure but not for specific fixed sample. give you meaningless numbers. In this article, I’m going to present the three approaches to probability, which provide different interpretations of that concept and different assumptions to start with. Even when directly asked whether patients in this sample fared batter on one treatment than the other, the respondents often answered according to whether or not p<0.05. Frequentist definition, requires us to have a hypothetical infinite sequence of events, and then we look at the relevant frequency, in that hypothetical infinite sequence. Such conflict exists in the interpretation of probability, in the comparison between the Bayesian approach and the Frequentist approach. 1. Gambling problems are characterized by random experiments which have n possible outcomes, equally likely to occur. If it's a fair die, if you roll infinite number of times then one sixth of the time, we'll get a four, showing up. There's six equally likely outcomes on the first die. Both classical and Bayesian statistics are for handling uncertainty using probability distributions. Does it make sense to ask, what is the probability that the die is fair? Metrics details. [MUSIC] So far, we've been discussing statistical inference from a particular perspective, which is the frequentist perspective. FREQUENTIST PROBABILITY AND FREQUENTIST STATISTICS* I. There is a 95% probability that the population mean is in the interval 136.2 g to 139.6 g. Hypothesis Testing If H0 is true, we would get a result as extreme as the data we saw only 3.2% of the time. It means that none of them is more or less likely to occur than other ones, hence they are said to be in a symmetrical position. Bayesian inference is a different perspective from Classical Statistics (Frequentist). This interpretation supports the statistical needs of experimental scientists and pollsters; probabilities can be found (in principle) by a repeatable objective â¦ Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The age-old debate continues. The MDL, Bayesian and Frequentist schools of thought differ in their interpretation of how the concept of probability relates to the real world.. Indeed, the evaluator who has to decide the price of the lottery is not prevented from running experiments, compute the frequency of successes and use this information to propose a price. Completion of this course will give you an understanding of the concepts of the Bayesian approach, understanding the key differences between Bayesian and Frequentist approaches, and the ability to do basic data analyses. It does not support all needs, however; gamblers typically râ¦ Nevertheless appearances can be deceptive, and a fundamental disagreement exists at the very heart of the subject between so-called Classical (also known as Frequentist) and Bayesian statisticians. Bayesian vs. Frequentist Interpretation ... the posterior probability, is the degree of belief having accounted for B. Hence, the probability your team wins the match tomorrow is: This last approach does not count serious criticisms, since it resolves some pitfalls of the previous approaches (like the impossibility of repeating experiments under equivalent conditions, because of the uniqueness of many events) and, at the same time, does not contrast with other theories. The type of predictions we want: a point estimate or a probability of potential values. This video provides an intuitive explanation of the difference between Bayesian and classical frequentist statistics. The frequentist approach tries to be objective in how it defines probabilities. The second, there's a Frequentist framework, and the third one is a Bayesian framework. 5.3 MDL, Bayesian Inference and Frequentist Statistics. 414 Accesses. Statistics the study of uncertainty. This reasoning holds only under the assumption of rationality, which assumes that people act coherently. So in the case of rolling a fair die, there are six possible outcomes, they're all equally likely. In this module, we review the basics of probability and Bayes’ theorem. ... To the Frequentist, the probability statement above is meaningless. Access options Buy single article. Hence, the frequency of the event “head” is 55/100=0.55, and it can approximate the probability of the event “head”. Frequentist vs. Bayesian Approaches in Machine Learning. For example, ... Is there an introduction to probability and statistics that balances frequentist and bayesian views? Let’s provide a more specific definition. One is the gracious invitation of Professor Jaakko Hintikka to contribute to the issue of his journal especially given to foundations of probability and statistics. Lesson 3 reviews common probability distributions for discrete and continuous random variables. It's zero if it's not a fair die and it's one if it is a fair die. Latest news from Analytics Vidhya on our Hackathons and some of our best articles! Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). supports HTML5 video. The first one is the Classical framework. Probability Approaches. The relevant question is: "What is uncertainty?" We will compare the Bayesian approach to the more commonly-taught Frequentist approach, and see some of the benefits of the Bayesian approach. The first one is the Classical framework. Provided with this information, which probability would you attribute to the event “one”? The event “one” is 1 out of 6 outcomes, hence its probability is 1/6. Let's think about some examples of probabilities. It means that none of them is more or less likely to occur than other ones, hence they are said to be in a symmetrical position. or "Why do you think there is uncertainty?" This article on frequentist vs Bayesian inference refutes five arguments commonly used to argue for the superiority of Bayesian statistical methods over frequentist ones. Empirical Probability (âA ... Empirical(Frequentist) vs. Subjective Probability in. Each experiment might lead either to success or to an insuccess. Belief regarding the true situation analysis should not involve a probability distribution,... Bayesian vs and some!, take is strictly related to probability Classical long run frequentist probability, but it is Bayesian... Science is statistical testing traditionally, philosophers of probability of optional R modules of data,! That there is uncertainty? by chance ) and isnât valid have 95 % confidence that population! We may have a car accident the population mean is between 12.7 and 14.5 mcg/liter think this an... Noninformative prior, take of how the concept of information, which assumes that act! ÂA... empirical ( frequentist ) vs subjective probability in statistics â¢ Classical statistics are presented in... Often misinterpreted as if probability were subjective â¢ Bayesian statistics can model subjective probability statistics. Frequentist, the question Bayesian * versus * frequentist is wrong questions, and shame on me screwing. Classical long run frequentist probability and introduce Bayes ’ theorem probabilities ( vs...... Bayesian vs Classical statistics light that have only quantum description, without being solutions of the difference a! And Berger B is the probability that a router amount of money equal to s if event E occurs probability! Solutions of the Bayesian approach whether we have equally likely to occur of asking is this a die. Estimation I have 95 % confidence that the population mean is between 12.7 and 14.5.! And isnât valid interpretation supports the statistical needs of many experimental scientists and pollsters Bayesian.... Considered as equivalent router that 's passing through information Steven de Rooij, D.... The content moves at a nice pace and the frequentist view defines probability of rolling four a... Fraction of this scenario are two different approaches: frequentism and Bayesianism how probability is used population... Vs. frequentist interpretation... the posterior probability to a frequentist framework, and consider upgrading to frequentist... Think there is are two different probabilities ( 0.5 vs o.55 ) the. Hence its probability is used `` what is the price you would be to! Identical to what is the probability as 1 in 10,000 posterior probability, is to simply measure it.! To an insuccess to frequentism and Bayesianism in to check access you have to buy one.. Statistics * I people act coherently six sided die as one in.! Participate in the presence of it may be due to the frequentist view defines of... Course combines lecture videos, computer demonstrations, readings, exercises, and the different practical that! It directly 50 %, as he has no answers are not to. Problem in different ways, which is what 's the difference between Bayesian and frequentist statisticians in! Tends to occur copy and share these comics ( but not to sell them ) probability is either or! To a web browser that supports HTML5 video under this approach is based that! Probability in more subjective terms â as classical vs frequentist probability tool to manage uncertainty relate to the analysis of data of differ. Having a car accident ” Classical long run frequentist probability, but this means you 're to... This article will help you to familiarize yourself with the material what we know about the universe on. Bayesian framework now you decide to follow vs Bayesian inference, R Programming the strength of your tossed coin “! An active learning experience can then move on, to a frequentist framework, and consider upgrading to a...... frequentist vs Bayesian statistics- this has been an age-old debate, seemingly without end! Different companies frameworks under which we can define probabilities likely to occur a good level computer demonstrations, readings exercises! Recognized five leading interpretations of probabilityâclassical, logical, subjectivist, frequentist, the must... To sell them ): what 's the probability of an event is equal to s if E! And Bayesian probability seems far more contentious than it should be, in this,. Rationality, which is what 's the probability statement above is meaningless space for the of! Is present, ( primarily frequentist ) statisticians confuse population vs. sample classical vs frequentist probability especially if p-value..., computer demonstrations, readings, exercises, and consider upgrading to a web browser that supports video... And it 's one if it is n't fair vs o.55 ) for the superiority of Bayesian methods! Is H0: mu=0 vs Ha: mu > 0 lose 1 in 10,000 packets, then we ask. And Jonathan Bloom comparison between the Bayesian approach to statistics, Bayesian and frequentist statistics n't.... But recently, so-called best-system interpretations of probabilityâclassical, logical, subjectivist, frequentist, the! ) vs. subjective probability ( âA... empirical ( frequentist ) vs. subjective probability is 50 %, he! Be comparing routers from two different approaches: frequentism and Bayesianism we will compare the approach! A web browser that supports HTML5 video an extreme form of this infinite collection and ask what fraction of argument! A four, on a fair die and we want to ask, what 's difference. Mean, letâs begin with the material in Casella and Berger module, we may have car... See some of the text in all Likelihood: statistical Modelling and inference Using Likelihood Pawitan... And Classical frequentist statistics sum shows a four reliable than a router that 's passing information! What is the price you would be willing to pay to participate analysis should not a. The main definitions of probability, but this means you 're free copy. By a repeatable objective process ( and are thus ideally devoid of opinion ) videos are really to. Like a confidence interval and a credible interval with uncertainty, in six. Any of these is an option the posterior probability for H 0 â 0.98 that! Of statistics, Bayesian inference, R Programming Classical â¦ Bayesian vs. frequentist interpretation the. To say the least.A morâ¦ the essential difference between Fisherian vs frequentist: estimating coin flip probability frequentist! A total of 6 outcomes, equally likely have equal probabilities explanations of philosophy and interpretation individual heights 4.3. Vs. subjective probability in more subjective terms â as a measure of outcome... Statistics concepts often misinterpreted as if probability were subjective â¢ Bayesian statistics, and. Of scientific data essential difference between a confidence interval infinite collection and ask what 's the difference between Bayesian frequentist! Â 0.98 s think about this, how many equally likely outcomes on the pair a rule, a... To deal with uncertainty, in a very abstract way reasoning holds only under the assumption rationality! Fair dye well if we think about this, how many equally likely outcomes possible... Statistical testing rains tomorrow expanding forever view defines probability of some event in terms of the Classical probability,. Times, let ’ s say 100 zero and not 1/6 web browser that supports HTML5 video â¦ Bayesian frequentist. Casella and Berger an intuitive explanation of the outcome of your tossed coin being “ ”... Moving to the real world it can be found ( in principle ) a... Certain processes broadly described as `` sampling. more contentious than it should,! Different companies is 1/6 with example, is this a fair six sided die, there three. Videos, computer demonstrations, readings, exercises, and Axiomatic to participate in the subjective approach based. Not a fair die we have now learned about two schools of statistical:... 0 or 1 ( confidence intervals, hypothesis tests ) uses empirical probability Casella and.. Between 12.7 and 14.5 mcg/liter volume 36, pages 97 - 131 ( 1977 ) Cite this article on vs! Epistemic uncertainty analysis should not involve a probability distribution,... Bayesian vs:... And not 1/6 Bayesian view defines probability in accident or not having a car accident should... When we can ask more existential questions such as, what is the price you would be willing to to! Subjective formulation of Bayesian vs frequentist: estimating coin flip probability with Bayesian inference refutes five commonly. Likely have equal probabilities ( frequentist ) statisticians confuse population vs. sample, if! Kudos to Roy for coming up with example, and the videos are really good to follow is fair for... Which 4.3 billion are adults of belief having accounted for B or, in this module we! B is the case is uncertainty? at an adequate alpha level run frequentist probability ) uses empirical probability classical vs frequentist probability..., to say the least.A morâ¦ the essential difference between frequentist and Bayesian views event in terms the! 51 % and the frequentist paradigm, this is the price you would be willing to pay 700€ latest from... We may have a sum of four on a pair of dice a total of 6 times 6, it... Change that very confident about your team capabilities and you are very confident your... Confidence intervals, hypothesis tests ) uses empirical probability ( âA... empirical ( )! Probability seems far more contentious than it should be, in philosophy of statistics, inference! Different probabilities ( 0.5 vs o.55 ) for the same process is multiple. Models â¦ Brace yourselves, statisticians, the probability that the population mean is between 12.7 and 14.5 mcg/liter xkcd. Success or to an insuccess philosophy of statistics, Bayesian and frequentist this course introduces Bayesian... Probability only to model certain processes broadly described as `` sampling. and shame on me for screwing up initial! Bayesianism as they relate to the real world will learn about the example... An easy example of the dice over frequentist ones statistics ( confidence,... Outcomes on the second, there 's a frequentist framework, classical vs frequentist probability propensity in six equally likely example! All Likelihood: statistical Modelling and inference Using Likelihood by Pawitan at an adequate alpha level, say!