Generally speaking, frequentist approaches posit that the world is one way (e.g., a parameter has one particular true value), and try to conduct experiments whose resulting conclusion -- no matter the true value of the parameter -- will be correct with at least some minimum probability.

As a result, to express uncertainty in our knowledge after an experiment, the frequentist approach uses a "confidence interval" -- a range of values designed to include the true value of the parameter with some minimum probability, say 95%. A frequentist will design the experiment and 95% confidence interval procedure so that out of every 100 experiments run start to finish, at least 95 of the resulting confidence intervals will be expected to include the true value of the parameter. The other 5 might be slightly wrong, or they might be complete nonsense -- formally speaking that's ok as far as the approach is concerned, as long as 95 out of 100 inferences are correct. (Of course we would prefer them to be slightly wrong, not total nonsense.)

Bayesian approaches formulate the problem differently. Instead of saying the parameter simply has one (unknown) true value, a Bayesian method says the parameter's value is fixed but has been chosen from some probability distribution -- known as the prior probability distribution. (Another way to say that is that before taking any measurements, the Bayesian assigns a probability distribution, which they call a belief state, on what the true value of the parameter happens to be.) This "prior" might be known (imagine trying to estimate the size of a truck, if we know the overall distribution of truck sizes from the DMV) or it might be an assumption drawn out of thin air. The Bayesian inference is simpler -- we collect some data, and then calculate the probability of different values of the parameter GIVEN the data. This new probability distribution is called the "a posteriori probability" or simply the "posterior." Bayesian approaches can summarize their uncertainty by giving a range of values on the posterior probability distribution that includes 95% of the probability -- this is called a "95% credibility interval."

A Bayesian partisan might criticize the frequentist confidence interval like this: "So what if 95 out of 100 experiments yield a confidence interval that includes the true value? I don't care about 99 experiments I DIDN'T DO; I care about this experiment I DID DO. Your rule allows 5 out of the 100 to be complete nonsense [negative values, impossible values] as long as the other 95 are correct; that's ridiculous."

A frequentist die-hard might criticize the Bayesian credibility interval like this: "So what if 95% of the posterior probability is included in this range? What if the true value is, say, 0.37? If it is, then your method, run start to finish, will be WRONG 75% of the time. Your response is, 'Oh well, that's ok because according to the prior it's very rare that the value is 0.37,' and that may be so, but I want a method that works for ANY possible value of the parameter. I don't care about 99 values of the parameter that IT DOESN'T HAVE; I care about the one true value IT DOES HAVE. Oh also, by the way, your answers are only correct if the prior is correct. If you just pull it out of thin air because it feels right, you can be way off."

In a sense both of these partisans are correct in their criticisms of each others' methods, but I would urge you to think mathematically about the distinction. There don't need to be Bayesians and frequentists any more than there are realnumberists and integeristos; there are different kinds of methods that apply math to calculate different things. This is a complex subject with a lot of sides to it, of which these examples are a tiny part -- books on Bayesian analysis could fill many bookshelves, not to mention classical statistics, which would fill a whole library.

------------

Here's an extended example that shows the difference precisely in a discrete example.

When I was a child my mother used to occasionally surprise me by ordering a jar of chocolate-chip cookies to be delivered by mail. The delivery company stocked four different kinds of cookie jars -- type A, type B, type C, and type D, and they were all on the same truck and you were never sure what type you would get. Each jar had exactly 100 cookies, but the feature that distinguished the different cookie jars was their respective distributions of chocolate chips per cookie. If you reached into a jar and took out a single cookie uniformly at random, these are the probability distributions you would get on the number of chips:

A type-A cookie jar, for example, has 70 cookies with two chips each, and no cookies with four chips or more! A type-D cookie jar has 70 cookies with one chip each. Notice how each vertical column is a probability mass function -- the conditional probability of the number of chips you'd get, given that the jar = A, or B, or C, or D, and each column sums to 100.

I used to love to play a game as soon as the deliveryman dropped off my new cookie jar. I'd pull one single cookie at random from the jar, count the chips on the cookie, and try to express my uncertainty -- at the 70% level -- of which jars it could be. Thus it's the identity of the jar (A, B, C or D) that is the

**value of the parameter**being estimated. The number of chips (0, 1, 2, 3 or 4) is the

**outcome**or the observation or the sample.

Originally I played this game using a frequentist, 70% confidence interval. Such an interval needs to make sure that

**no matter**the true value of the parameter, meaning no matter which cookie jar I got, the interval would cover that true value with at least 70% probability.

An interval, of course, is a function that relates an outcome (a row) to a set of values of the parameter (a set of columns). But to

*construct*the confidence interval and guarantee 70% coverage, we need to work "vertically" -- looking at each column in turn, and making sure that 70% of the probability mass function is covered so that 70% of the time, that column's identity will be part of the interval that results. Remember that it's the vertical columns that form a p.m.f.

So after doing that procedure, I ended up with these intervals:

For example, if the number of chips on the cookie I draw is 1, my confidence interval will be {B,C,D}. If the number is 4, my confidence interval will be {B,C}. Notice that since each column sums to 70% or greater, then no matter which column we are truly in (no matter which jar the deliveryman dropped off), the interval resulting from this procedure will include the correct jar with at least 70% probability.

Notice also that the procedure I followed in constructing the intervals had some discretion. In the column for type-B, I could have just as easily made sure that the intervals that included B would be 0,1,2,3 instead of 1,2,3,4. That would have resulted in 75% coverage for type-B jars (12+19+24+20), still meeting the lower bound of 70%.

My sister Bayesia thought this approach was crazy, though. "You have to consider the deliverman as part of the system," she said. "Let's treat the identity of the jar as a random variable itself, and let's

*assume*that the deliverman chooses among them uniformly -- meaning he has all four on his truck, and when he gets to our house he picks one at random, each with uniform probability."

"With that assumption, now let's look at the joint probabilities of the whole event -- the jar type

**and**the number of chips you draw from your first cookie," she said, drawing the following table:

Notice that the whole table is now a probability mass function -- meaning the whole table sums to 100%.

"Ok," I said, "where are you headed with this?"

"You've been looking at the conditional probability of the number of chips, given the jar," said Bayesia. "That's all wrong! What you really care about is the conditional probability of which jar it is, given the number of chips on the cookie! Your 70% interval should simply include the list jars that, in total, have 70% probability of being the true jar. Isn't that a lot simpler and more intuitive?"

"Sure, but how do we calculate that?" I asked.

"Let's say we

**know**that you got 3 chips. Then we can ignore all the other rows in the table, and simply treat that row as a probability mass function. We'll need to scale up the probabilities proportionately so each row sums to 100, though."

She did:

"Notice how each row is now a p.m.f., and sums to 100%. We've flipped the conditional probability from what you started with -- now it's the probability of the man having dropped off a certain jar, given the number of chips on the first cookie."

"Interesting," I said. "So now we just circle enough jars in each row to get up to 70% probability?" We did just that, making these credibility intervals:

Each interval includes a set of jars that,

*a posteriori*, sum to 70% probability of being the true jar."Well, hang on," I said. "I'm not convinced. Let's put the two kinds of intervals side-by-side and compare them for coverage and, assuming that the deliveryman picks each kind of jar with equal probability, credibility."

Here they are:

**Confidence intervals:**

"See how crazy your confidence intervals are?" said Bayesia. "You don't even have a sensible answer when you draw a cookie with zero chips! You just say it's the empty interval. But that's obviously wrong -- it has to be one of the four types of jars. How can you live with yourself, stating an interval at the end of the day when you

**know the interval is wrong?**And ditto when you pull a cookie with 3 chips -- your interval is only correct 41% of the time. Calling this a '70%' confidence interval is bullshit.""Well, hey," I replied. "It's correct 70% of the time, no matter which jar the deliveryman dropped off. That's a lot more than you can say about your credibility intervals. What if the jar is type B? Then your interval will be wrong 80% of the time, and only correct 20% of the time!"

"This seems like a big problem," I continued, "because your mistakes will be correlated with the type of jar. If you send out 100 'Bayesian' robots to assess what type of jar you have, each robot sampling one cookie, you're telling me that on type-B days, you will expect 80 of the robots to get the wrong answer, each having >73% belief in its incorrect conclusion! That's troublesome, especially if you want most of the robots to agree on the right answer."

"PLUS we had to make this assumption that the deliveryman behaves uniformly and selects each type of jar at random," I said. "Where did that come from? What if it's wrong? You haven't talked to him; you haven't interviewed him. Yet all your statements of

*a posteriori*probability rest on this statement about his behavior. I didn't have to make any such assumptions, and my interval meets its criterion even in the worst case."

"It's true that my credibility interval does perform poorly on type-B jars," Bayesia said. "But so what? Type B jars happen only 25% of the time. It's balanced out by my good coverage of type A, C, and D jars. And I never publish nonsense."

"It's true that my confidence interval does perform poorly when I've drawn a cookie with zero chips," I said. "But so what? Chipless cookies happen, at most, 27% of the time in the worst case (a type-D jar). I can afford to give nonsense for this outcome because NO jar will result in a wrong answer more than 30% of the time."

"The column sums matter," I said.

"The row sums matter," Bayesia said.

"I can see we're at an impasse," I said. "We're both correct in the mathematical statements we're making, but we disagree about the appropriate way to quantify uncertainty."

"That's true," said my sister. "Want a cookie?"

I very much like your story about cookies. But I don't think Bayesia has any reason to claim "Type B jars happen only 25% of the time." I think a fairer characterization of the Bayesian position would be "I have no reason to think Type B jars are especially common, so I don't think poor performance on Type B is that big of a problem." Regardless, nice post.

ReplyDeleteP.S. Did you happen to post a version of this on a HackerNews comment many years ago? Or maybe someone else did? I have a faint recollection of this 4x4 confidence interval story to illustrate the difference between frequentists and Bayesians.

Assuming uniform apriori distribution, Type B jars happen 25% of the time!

DeleteUniform over what space of models? That is the key question.

DeleteThe story is great at explaining the conceptual differences between the two approaches. However, the way the performances are currently compared leaves the impression that it is a matter of taste which method to prefer. I think this is misleading. You never explicitly say how one should evaluate the performance of a given method but you create the impression that it is simply a matter of how often the true value is contained in the interval. However, by that criterium one can have a perfect method: simply always predict the interval that contains all jars. This is guaranteed to be true 100% of the time. But obviously such a prediction would be useless. The value of a prediction is given by how much it narrows down the possibilities. So one inherently is trading risk of a wrong prediction against making more specific predictions.

ReplyDeleteOnce one realizes that, it becomes rather telling that the Bayesian method not only makes a prediction for every case (i.e. it does not refuse to make a prediction when one got zero chips) but the intervals are significantly shorter on average as well.

If one wants to compare the methods in a rigorous way one would have to specify what the value V(n) is of a prediction that narrows the number of possibilities down from 4 to n if it turns out to be correct, and what the cost C(n) is, if it is incorrect.

Once one does that, you will find that the optimal intervals are the ones that maximize the Bayesian posterior expectation of V-C.

This comment has been removed by the author.

ReplyDeleteHere is one more helpful blog with essay writing advises.

ReplyDeleteThe two previous comments by "Ralphes Bushman" look like spam. Do not click the links.

ReplyDeleteThe 2017 Pro Bowl is the National Football League's all-star game for the 2016 season which will be played at Camping World Stadium in Orlando, Florida on January 29, 2017.

ReplyDeletePro Bowl 2017

Pro Bowl 2017 Live

Pro Bowl 2017 Live Stream

Pro Bowl 2017

Pro Bowl 2017 Live

Royal Rumble 2017

Super Bowl 2017

ReplyDeleteNFL Super Bowl 2017

2017 Super Bowl

2017 NFL Super Bowl

Super Bowl 2017 Game

Super Bowl 51

Super Bowl 51 Live

Super Bowl 2017 Live

ReplyDeleteNFL Super Bowl 2017 Live

Super Bowl 2017 Live Stream

Patriots vs Falcons 2017

Falcons vs Patriots

Patriots vs Falcons

Falcons vs Patriots 2017

2017 Grammy Awards: Complete list of nominees, GRAMMY Awards Start on February 12, 2017 - Watch Live on CBS

ReplyDeleteGrammys

Grammy

Grammy 2017

2017 Grammy

Grammys 2017

2017 Grammys

Grammys live

Grammys live Stream

Watch Grammys

Grammys Awards

Grammy Awards

Grammy Awards

Grammy Awards 2017

59th Grammy Awards

59th Annual Grammy Awards

Grammys Awards 2017

Alright, but what about the other movies? There are nine best picture contenders, you know, and many other acting nominees besides Gosling and Stone

ReplyDeleteOscar 2017

We’re nearly there, movie fans — awards season is drawing to a close. In less than two weeks, the annual parade of galas and ceremonies honoring the films of the

Oscar 2017 Live

RONDA ROUSEY was battered by Amanda Nunes in her long-awaited UFC comeback bout - here are some of the best pictures from the fight. Born in Sweden, Hermansson (14-3) looks to bounce back from his first UFC loss, a submission defeat to Cezar Ferreira that snapped .

ReplyDeleteUFC 209

UFC 209 Live Stream , UFC 209 Live

UFC 209 Fight , UFC 209 Fight Card

UFC lightweight Marc Diakiese is putting out a strong message of tolerance and acceptance, become the first fighter for the world's largest MMA. CONOR McGREGOR has been pictured with Manchester United star Wayne Rooney as talks of a superfight with Floyd Mayweather rumble on.

UFC 209

UFC 209 Live , UFC 209 Fight Card

UFC 209 Live Stream , UFC 209 Fight

Westbrook has pushed the envelope of what any player has done; using 45 percent of his team's possessions is the highest I've ever

ReplyDeleteMarch Madness

March Madness Live

March Madness Live Stream

March Madness 2017

March Madness 2017 Live

ncaa March Madness

ncaa march madness live

ncaa tournament

March Madness Bracket

ncaa final four

een. And you don't see players working that much offensively who maintain their defense as much as he does. Usually guys who shoot a ton

Idaho: Although that Tar Heels displaced in the ACC competition semifinals, on the list of Fight it out in the rules meant for most of the initial 50 percent. Justin Fitzgibbons, who's got produced a good circumference sport, in addition to a foul-free Joel Acai berry II are definitely the first considerations.

ReplyDeleteMarch Madness

March Madness Live

March Madness Live Stream

March Madness 2017

Absolutely, we all know Level Few has never built one final Five. The following company, stored by means of proficient scorers that will as well locking mechanism lower attackers, will be her first

March Madness 2017 Live

ncaa March Madness

ncaa march madness live

ncaa tournament

March Madness Bracket

ncaa final four

Whether you’re nevertheless over the fence this 8-9 sport, or even are trying to find your consensus feeling onto your Finalized Several choices, we’ve gotten everyone dealt with using a rating of all 68 clubs inside competition. Anybody can with assurance generate for you to decide on hundreds of mid-major darlings you’ve recently been eyeing with regard to disappointed likely.

March Madness

March Madness Live

March Madness Live Stream

March Madness 2017

March Madness 2017 Live

ncaa March Madness

ncaa march madness live

ncaa tournament

March Madness Bracket

ncaa final four

Villanova: The most notable entire seed. Despite the team’s position inside the toughest location (the Distance, with matchups next to Fight it out and additionally SMU looming), Villanova’s three-man offensive major from Jalen Brunson, Josh Hart and additionally Kris Jenkins offers the possibilities to be able to try for the reason that national champs.

orders before it was located 43 days after going MIA after one of the greatest football finishes ever seen. To find that jersey took a similar effort from all agencies involved.

ReplyDeleteMarch Madness

March Madness Live

March Madness Live Stream

March Madness 2017

March Madness 2017 Live

ncaa March Madness

ncaa march madness live

ncaa tournament

March Madness Bracket

ncaa final four

The search also reached the desk of assistant U.S. attorney John Durham of New Haven, Connecticut, sources told ESPN. Durham was critical to the

The Masters Tournament, also known as The Masters or The US Masters, is one of the four major championships in professional golf. The Masters is scheduled for the first full week of April, and it is the first of the majors to be played each year.

ReplyDeleteThe masters

The masters 2017

The masters Golf

The masters Golf 2017

The masters Live

The masters Live Stream

Masters golf result

Masters results

Masters Golf Leaderboard

Master Leaderboard

The masters

The masters 2017

The masters Golf

The masters Golf 2017

The masters Live

The masters Live Stream

Masters golf result

Masters results

Masters Golf Leaderboard

Master Leaderboard

Masters Tournament 2017 live coverage from Augusta National Golf Club at CBSSports.com. Watch the tournament live, choose cameras and get live stats.

ReplyDeleteUFC 210UFC 210 Live210 UFCWatch UFC 210UFC 210 FightUFC 210 Live StreamWatch UFC 210 LiveCormier vs Johnson 2UFC 210 Cormier vs Johnson LiveUFC 210 FightUFC 210 Fight CardThis time, we head to Birmingham, Alabama to check out UFC 10 from July 1996, which brings back the eight-man one-night tournament format. Barely a decade ago, the Ultimate Fighting Championship was looked at as little more than a freak show a human cockfight that drew the attention.

ReplyDeleteUFC 210

UFC 210 Live , UFC 210 Fight Card , UFC 210 Fight , UFC 210 Card , UFC 210 Live Stream , UFC 210 PPV

UFC 10 Review: Mark Coleman Enters The Octagon For. Joel AbrahamContributor IIIDecember 24, 2010. I am still having nightmares about UFC 10. FS1 presents the Ultimate Fighting Championship's top pound-for-pound fighters across two weeks of prime time action with the UFC 10 Day.

UFC 210

UFC 210 Live , UFC 210 Fight Card , UFC 210 Fight , UFC 210 Card , UFC 210 Live Stream , UFC 210 PPV

ReplyDeleteUFC 210

UFC 210 Live

UFC 210 Live Stream

UFC 210 Live online

Watch UFC 210

UFC 210 Live PPV

Watch UFC 210 Online

Cormier vs Johnson 2

Cormier vs Johnson 2 Live

Cormier vs Johnson 2 Live Stream

The IPL 2017 is the most watched Cricket league in the world. It is a tournament where renowned international cricketers come together on one stage & budding Indian players are groomed under their guidance. IPL 2017 Live Scores is where Talent Meets Opportunity.

ReplyDeleteJoshua vs Klitschko is a professional boxing match to be contested between Anthony Joshua vs Klitschko Live. Joshua vs Klitschko Live Stream is a professional boxing match to be contested between Joshua vs Klitschko Fight. Watch Joshua vs Klitschko event will take place on 29 April 2017 at Wembley Stadium in London, England, with Joshua's IBF and the vacant WBA (Super) and IBO heavyweight titles on the line. The Klitschko vs Joshua Ukrainian has not fought since then and at 41-years-old, many are expecting him to lose back-to-back fights for the first time in his career.

ReplyDelete

ReplyDeleteThe Players Championship 2017

The Players Championship 2017 Live

The Players Championship 2017 Live Stream

Players Championship 2017

Players Championship 2017 Live

Players Championship 2017 Live Stream

2017 Players Championship

2017 Players Championship Live

2017 Players Championship Live Stream

UFC 211

ReplyDeleteUFC 211 Live

UFC 211 Live stream

UFC 211 Live Online

Watch UFC 211

UFC 211 Live Streaming

How to watch UFC 211

UFC 211 Live Free

Miocic vs dos Santos 2

Miocic vs dos Santos 2 Live

Players Championship 2017

Players Championship 2017 Live

Players Championship 2017 Live Stream

Players Championship 2017 Live online

http://opticsforhunter.kinja.com/

ReplyDeletehttp://opticsforhunter.kinja.com/

http://opticsforhunter.kinja.com/

http://opticsforhunter.kinja.com/

http://opticsforhunter.kinja.com/

Cardinals vs Cowboys Live

ReplyDeleteCardinals vs Cowboys Live Stream

Cowboys vs Cardinals Live

Cowboys vs Cardinals Live Stream

NFL Hall of Fame 2017

Mayweather Vs Mcgregor

ReplyDeleteMcgregor vs Mayweather

Mayweather Mcgregor

Mcgregor Mayweather

Mayweather Vs Mcgregor Fight

Mcgregor vs Mayweather Fight

Mayweather Mcgregor Fight

Mcgregor Mayweather Fight

Mayweather Vs Mcgregor live

Mcgregor vs Mayweather live

Mayweather Vs Mcgregor

Mcgregor vs Mayweather

Mayweather Mcgregor

Mcgregor Mayweather

Mayweather Vs Mcgregor live

Mcgregor vs Mayweather live

Mayweather Vs Mcgregor live Stream

Mayweather Vs Mcgregor Time

Mcgregor vs Mayweather Time

Mayweather Vs Mcgregor Fight

Black Friday

ReplyDeleteBlack Friday Ads

Black Friday Deals

Black Friday Sales

Black Friday 2017

Amazon Black Friday

Amazon Black Friday 2017

Black Friday Deals 2017

Black Friday Themes

Black Friday Web Hosting

Best Black Friday

Black Friday

Black Friday Ads

Black Friday Deals

Black Friday Sales

Black Friday 2017

Amazon Black Friday

Amazon Black Friday 2017

Black Friday Deals 2017

Black Friday Themes

Best Black Friday

UFC 218

ReplyDeleteUFC 218 Live

UFC 218 Live Stream

Watch UFC 218

UFC 218 Fight

UFC 218 Fight Card

Holloway vs Aldo 2

Miami vs Clemson

Clemson vs Miami

Wisconsin vs Ohio State

Ohio State vs Wisconsin

Georgia vs Auburn

Auburn vs Georgia

TCU vs Oklahoma

Oklahoma vs TCU