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01.09.2014
Suppose we wish to estimate the mean systolic blood pressure, body mass index, total cholesterol level or white blood cell count in a single target population. Descriptive statistics on variables measured in a sample of a n=3,539 participants attending the 7th examination of the offspring in the Framingham Heart Study are shown below. The table below shows data on a subsample of n=10 participants in the 7th examination of the Framingham offspring Study. Suppose we compute a 95% confidence interval for the true systolic blood pressure using data in the subsample. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Joris and Srikant's exchange here got me wondering (again) if my internal explanations for the the difference between confidence intervals and credible intervals were the correct ones.
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 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? 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?
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 -- as Srikant explains.
Here's an extended example from that talk 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. A type-A cookie jar, for example, has 70 cookies with two chips each, and no cookies with four chips or more! 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). Notice also that the procedure I followed in constructing the intervals had some discretion. Notice that the whole table is now a probability mass function -- meaning the whole table sums to 100%. Each interval includes a set of jars that, a posteriori, sum to 70% probability of being the true jar. Your inference problem is: What values of $\theta$ are reasonable given the observed data $x$ ? The above understanding leads to the following methodology to assess where the true parameter is in relation to your estimate. In this paradigm, it is meaningless to speak about the probability that a true value is less than or greater than some value as the true value is not a random variable.
In contrast to the classical approach, in the bayesian approach we assume that the true value is a random variable. Credible intervals capture our current uncertainty in the location of the parameter values and thus can be interpreted as probabilistic statement about the parameter. A 95% confidence interval by definition covers the true parameter value in 95% of the cases, as you indicated correctly. Confidence interval (CI) is a concept based on the classical definition of probability (also called the "Frequentist definition") that probability is like proportion and is based on the axiomatic system of Kolmogrov (and others). Credible intervals (Highest Posterior Density, HPD) can be considered to have its roots in decision theory, based on the works of Wald and de Finetti (and extended a lot by others). As people in this thread have done a great job in giving examples and the difference of hypotheses in the Bayesian and frequentist case, I will just stress on a few important points. In general CIs are NOT coherent (will be explained later) where as HPDs are coherent(due to their roots in decision theory). As CIs don't condition on the observed data (also called "Conditionality Principle" CP), there can be paradoxical examples. Kiefer did some foundational work on CI in the late 1970s, but his extensions haven't become popular. CIs can't be interpreted as probability and they don't tell anything about the unkown parameter GIVEN the observed data. HPDs are probabilistic intervals based on the posterior distribution of the unknown parameter and have a probability based interpretation based on the given data.
Frequentist property (repeated sampling) property is a desirable property and HPDs (with appropriate priors) and CI both have them. You can see this quite clearly, for the type A and type D jars make "definite predictions" so to speak (for 0-1 and 2-3 chips respectively), whereas type B and C jars basically give a uniform distribution of chips. And from the "practical" viewpoint, type B and C would require an enormous sample to be able to distinguish between them. Another point I would like to stress is that the Bayesian is not saying that "the parameter is random" by assigning a probability distribution. A confidence interval is based on the "randomness" or variation which exists in the different possible samples. Note that this crazy behavior is certainly not what a proponent of confidence intervals would actually do; but it does demonstrate a weakness in the philosophy underlying the method in a particular case. Estimates of economic and fiscal variables over the forward estimates period are subject to inherent uncertainties, which generally tend to increase as the forecast horizon lengthens. For real and nominal GDP forecasts, confidence intervals could be presented around forecasts of annual growth rates, average annualised growth rates or cumulative growth rates.

The uncertainty around nominal GDP is larger, reflecting uncertainty about the outlook for real GDP and uncertainty about the outlook for prices or the GDP deflator. It should be noted that excluding historical variations due to policy decisions does not exclude cases that are classified in budget documentation as parameter and other variations, but have more in common with decisions of government. Chart B3 suggests that there is notable uncertainty around receipts forecasts and that this uncertainty increases over the estimates period. Chart B4 suggests that there is moderate uncertainty around payments forecasts and that this uncertainty exhibits apparent but contained growth over the estimates period.
Chart B5 suggests that there is notable uncertainty around the underlying cash balance forecasts and that this uncertainty exhibits pronounced growth over the estimates period. 1 The 2012 Review of Treasury Macroeconomic and Revenue Forecasting found that the official macroeconomic and tax revenue forecasting performance is comparable with or better than that of other forecasters, suggesting that the uncertainty around the forecasts is similar to or smaller than those of other forecasters. 2 The confidence intervals around the fiscal forecasts are based on GDP outcomes, rather than GDP forecasts, as discussed in Treasury Working Paper 2013-04: Estimates of uncertainty around budget forecasts which found that forecast errors for GDP and receipts (in particular) are highly correlated. 3 The impacts of past policy decisions on historical public debt interest through time cannot be readily identified or estimated.
We select a sample and compute descriptive statistics including the sample size (n), the sample mean, and the sample standard deviation (s). The margin of error is very small (the confidence interval is narrow), because the sample size is large. Because the sample size is small, we must now use the confidence interval formula that involves t rather than Z. 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. 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.
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. 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. 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. 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. 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. You have a data generating process that describes how $x$ is generated conditional on $\theta$. Since, the true value is unknown but fixed, the true value is either in the interval or outside the interval.
Thus, we capture the our uncertainty about the true parameter value by a imposing a prior distribution on the true parameter vector (say $f(\theta)$). However, since under this paradigm, the true parameter vector is a random variable, we also want to know the extent of uncertainty we have in our point estimate. Thus, they cannot be interpreted as a probabilistic statement about the true parameter values. Yes, for any value of the parameter, there will be >95% probability that the resulting interval will cover the true value. My point is that it's possible to be in situations, after you take the sample, where you can prove with absolute certainty that your interval is wrong -- that it does not cover the true value. Coherence (as I would explain to my grand mom) means: given a betting problem on a parameter value, if a classical statistician (frequentist) bets on CI and a bayesian bets on HPDs, the frequentist IS BOUND to lose (excluding the trivial case when HPD = CI). Fisher was a big supporter of CP and also found a lot of paradoxical examples when this was NOT followed (as in the case of CI). But SP and CP together imply the Likelihood Principle (LP) (cf Birnbaum, JASA, 1962) ie given CP and SP, one must ignore the sample space and look at the likelihood function only. A good source of reference is Berger ("Could Fisher, Neyman and Jeffreys agree about testing of hypotheses", Stat Sci, 2003).
The CONFIDENCE INTERVAL (perhaps a misnomer) is interpreted as: if the experiment that generated the random sample x were repeated many times, 95% (or other) of such intervals constructed from those random samples would contain the true value of the parameter.
That's the problem with frequentist statistics and the main thing Bayesian statistics has going for it.
The prior P(?) is inherently subjective, but that is the price of knowledge about the Universe - in a very profound sense. While all three measures have merit, a key role of GDP forecasts is as an input for producing revenue and expenses forecasts. These charts show confidence intervals around the forecasts for receipts (excluding GST and including Future Fund earnings), payments (excluding GST) and the underlying cash balance (which excludes Future Fund earnings). To account for this, confidence intervals constructed around the fiscal variables exclude historical variations caused by policy decisions. For example, specific decisions to re‐profile spending due to changes in timing of projects are captured for reporting purposes as parameter and other variations. As a result, GST data have been removed from historical receipts and payments data to abstract from any error associated with this change in accounting treatment. The formulas for confidence intervals for the population mean depend on the sample size and are given below. 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.

Each jar had exactly 100 cookies, but the feature that distinguished the different cookie jars was their respective distributions of chocolate chips per cookie. 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. Thus it's the identity of the jar (A, B, C or D) that is the value of the parameter being estimated. 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. That would have resulted in 75% coverage for type-B jars (12+19+24+20), still meeting the lower bound of 70%.
Given this assumption, you use the data $x$ to get to an estimate of $\theta$ (say, $\hat{\theta}$). In contrast, your estimate is a random variable as it depends on your data $x$ which was generated from your data generating process. The confidence interval then is a statement about the likelihood that the interval we obtain actually has the true parameter value. You can sometimes say something about the chance that the parameter is larger or smaller than any of the boundaries, based on the assumptions you make when constructing the interval (pretty often the normal distribution of your estimate). That doesn't mean that after taking a particular observation and calculating the interval, there still is 95% conditional probability given the data that THAT interval covers the true value. Every confidence interval comes from a different random sample, so the chance that your sample is drawn such that the 95%CI on which it is based does not cover the true parameter value, is only 5%, regardless of the data. In short, if you want to summarize the findings of your experiement as a probability based on the data, the probability HAS to be a posterior probability (based on a prior).
Thus, we only need to look at the given data and NOT at the whole sample space (looking at whole sample space is in a way similar to repeated sampling).
For this purpose, the average annualised GDP growth rate or cumulative GDP growth rate is the more relevant summary statistic, since the level of GDP depends on cumulative growth over time. Similarly, new and often substantial spending decisions to provide assistance for the impacts of natural disasters are covered under the Natural Disaster Relief and Recovery Arrangements and are captured for budget reporting purposes as parameter and other variations.
The Bayesian inference is simpler -- we collect some data, and then calculate the probability of different values of the parameter GIVEN the data. 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. Once you have your estimate you want to assess where the true value is in relation to your estimate.
There is a theorem (cf Heath and Sudderth, Annals of Statistics, 1978) which (roughly) states: Assignment of probability to $\theta$ based on data will not make a sure loser if and only if it is obtained in a bayesian way. In his view p-values were based on the observed data (much can be said about p-values, but that is not the focus here). The average annualised growth rate is reported as it captures the effects of cumulative growth, while still giving a sense of what the annual growth rate would be. The treatment of these spending decisions contributes to the size of the confidence intervals around payments. 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. Efron and Hinkley, AS, 1978) which measure the information about the data from a frequentist perspective. The coverage is unchanged as all intervals contained both B and C or neither, so it is still subject to the criticisms in Keith's response - 59% and 0% for 3 and 0 chips observed. Confidence intervals are based on the root mean square errors (RMSEs) of Budget forecasts from 1998‑99 onwards, with outcomes based on December quarter 2013 National Accounts data. Further uncertainty from this source can be expected over the forecast period as provisions for impacts of future natural disasters are not included in estimates beyond the budget year.
Isn't it better (or sensible) to use the variance of sample mean estimator as $0.001\sigma^2$ (conditional variance) instead of the actual variance of the estimator, which is HUGE!! The interpretation of this CI is: if we sample repeatedly, we will get different ${\bar x}$ and 99% (at least) times it will contain true $\theta$, BUT (the elephant in the room) for a GIVEN data, we DON'T know the probability that CI will contain true $\theta$.
Fisher extended the strategy of conditioning on the ancillary statistic to a general theory called Fiducial Inference (also called his greatest failure, cf Zabell, Stat. The amount of information in the data is a bayesian concept (and hence related to HPD), instead of CI.
For example, in the basic linear models case, some distributions can be obtained by using a uniform prior (which some people called diffuse), BUT it DOESN'T mean that uniform distribution can be regarded as a LOW INFORMATION PRIOR.
In general, NON-INFORMATIVE(Objective) prior doesn't mean it has low information about the parameter. This is a simple illustration of CP when we use the variance as $0.001\sigma^2$ when $n=1000$. Fisher was trying to find a way different from both the classical statistics (of Neyman School) and the bayesian school (hence the famous adage from Savage: "Fisher wanted to make a Bayesian omelette (ie using CP) without breaking the Bayesian eggs").
This directly relates to CI as they involve the variance which should not be conditioned on $n$, ie we will end up using the larger variance, hence over conservative. Folklore (no proof) says: Fisher in his debates attacked Neyman (for Type I and Type II error and CI) by calling him a Quality Control guy rather than a Scientist, as Neyman's methods didn't condition on the observed data, instead looked at all possible repetitions.

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