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It is often the case that a confidence interval with 95% coverage is very similar to a credible interval that contains 95% of the posterior density. This happens when the prior is uniform or near uniform in the latter case. Thus a confidence interval can often be used to approximate a credible interval and vice versa. Importantly, we can conclude from this that the much maligned misinterpretation of a confidence interval as a credible interval has little to no practical importance for many simple use cases.

There are a number of examples out there of cases where this does not happen, however they all seem to be cherrypicked by proponents of Bayesian stats in an attempt to prove there is something wrong with the frequentist approach. In these examples, we see the confidence interval contains impossible values, etc which is supposed to show that they are nonsense.

I don't want to go back over those examples, or a philosophical discussion of Bayesian vs Frequentist.

I am just looking for examples of the opposite. Are there any cases where the confidence and credible intervals are substantially different, and the interval provided by the confidence procedure is clearly superior?

To clarify: This is about the situation when the credible interval is usually expected to coincide with the corresponding confidence interval, ie when using flat, uniform, etc priors. I am not interested in the case where someone chooses an arbitrarily bad prior.

EDIT: In response to @JaeHyeok Shin's answer below, I must disagree that his example uses the correct likelihood. I used approximate bayesian computation to estimate the correct posterior distribution for theta below in R:

### Methods ###
# Packages
require(HDInterval)

# Define the likelihood
like <- function(k = 1.2, theta = 0, n_print = 1e5){
  x    = NULL
  rule = FALSE
  while(!rule){
    x     = c(x, rnorm(1, theta, 1))
    n     = length(x)
    x_bar = mean(x)

    rule = sqrt(n)*abs(x_bar) > k

    if(n %% n_print == 0){ print(c(n, sqrt(n)*abs(x_bar))) }
  }
  return(x)
}

# Plot results
plot_res <- function(chain, i){
    par(mfrow = c(2, 1))
    plot(chain[1:i, 1], type = "l", ylab = "Theta", panel.first = grid())
    hist(chain[1:i, 1], breaks = 20, col = "Grey", main = "", xlab = "Theta")
}


### Generate target data ### 
set.seed(0123)
X = like(theta = 0)
m = mean(X)


### Get posterior estimate of theta via ABC ###
tol   = list(m = 1)
nBurn = 1e3
nStep = 1e4


# Initialize MCMC chain
chain           = as.data.frame(matrix(nrow = nStep, ncol = 2))
colnames(chain) = c("theta", "mean")
chain$theta[1]  = rnorm(1, 0, 10)

# Run ABC
for(i in 2:nStep){
  theta = rnorm(1, chain[i - 1, 1], 10)
  prop  = like(theta = theta)

  m_prop = mean(prop)


  if(abs(m_prop - m) < tol$m){
    chain[i,] = c(theta, m_prop)
  }else{
    chain[i, ] = chain[i - 1, ]
  }
  if(i %% 100 == 0){ 
    print(paste0(i, "/", nStep)) 
    plot_res(chain, i)
  }
}

# Remove burn-in
chain = chain[-(1:nBurn), ]

# Results
plot_res(chain, nrow(chain))
as.numeric(hdi(chain[, 1], credMass = 0.95))

This is the 95% credible interval:

> as.numeric(hdi(chain[, 1], credMass = 0.95))
[1] -1.400304  1.527371

enter image description here

EDIT #2:

Here is an update after @JaeHyeok Shin's comments. I'm trying to keep it as simple as possible but the script got a bit more complicated. Main changes:

  1. Now using a tolerance of 0.001 for the mean (it was 1)
  2. Increased number of steps to 500k to account for smaller tolerance
  3. Decreased the sd of the proposal distribution to 1 to account for smaller tolerance (it was 10)
  4. Added the simple rnorm likelihood with n = 2k for comparison
  5. Added the sample size (n) as a summary statistic, set tolerance to 0.5*n_target

Here is the code:

### Methods ###
# Packages
require(HDInterval)

# Define the likelihood
like <- function(k = 1.3, theta = 0, n_print = 1e5, n_max = Inf){
  x    = NULL
  rule = FALSE
  while(!rule){
    x     = c(x, rnorm(1, theta, 1))
    n     = length(x)
    x_bar = mean(x)
    rule  = sqrt(n)*abs(x_bar) > k
    if(!rule){
     rule = ifelse(n > n_max, TRUE, FALSE)
    }

    if(n %% n_print == 0){ print(c(n, sqrt(n)*abs(x_bar))) }
  }
  return(x)
}


# Define the likelihood 2
like2 <- function(theta = 0, n){
  x = rnorm(n, theta, 1)
  return(x)
}



# Plot results
plot_res <- function(chain, chain2, i, main = ""){
    par(mfrow = c(2, 2))
    plot(chain[1:i, 1],  type = "l", ylab = "Theta", main = "Chain 1", panel.first = grid())
    hist(chain[1:i, 1],  breaks = 20, col = "Grey", main = main, xlab = "Theta")
    plot(chain2[1:i, 1], type = "l", ylab = "Theta", main = "Chain 2", panel.first = grid())
    hist(chain2[1:i, 1], breaks = 20, col = "Grey", main = main, xlab = "Theta")
}


### Generate target data ### 
set.seed(01234)
X    = like(theta = 0, n_print = 1e5, n_max = 1e15)
m    = mean(X)
n    = length(X)
main = c(paste0("target mean = ", round(m, 3)), paste0("target n = ", n))



### Get posterior estimate of theta via ABC ###
tol   = list(m = .001, n = .5*n)
nBurn = 1e3
nStep = 5e5

# Initialize MCMC chain
chain           = chain2 = as.data.frame(matrix(nrow = nStep, ncol = 2))
colnames(chain) = colnames(chain2) = c("theta", "mean")
chain$theta[1]  = chain2$theta[1]  = rnorm(1, 0, 1)

# Run ABC
for(i in 2:nStep){
  # Chain 1
  theta1 = rnorm(1, chain[i - 1, 1], 1)
  prop   = like(theta = theta1, n_max = n*(1 + tol$n))
  m_prop = mean(prop)
  n_prop = length(prop)
  if(abs(m_prop - m) < tol$m &&
     abs(n_prop - n) < tol$n){
    chain[i,] = c(theta1, m_prop)
  }else{
    chain[i, ] = chain[i - 1, ]
  }

  # Chain 2
  theta2  = rnorm(1, chain2[i - 1, 1], 1)
  prop2   = like2(theta = theta2, n = 2000)
  m_prop2 = mean(prop2)
  if(abs(m_prop2 - m) < tol$m){
    chain2[i,] = c(theta2, m_prop2)
  }else{
    chain2[i, ] = chain2[i - 1, ]
  }

  if(i %% 1e3 == 0){ 
    print(paste0(i, "/", nStep)) 
    plot_res(chain, chain2, i, main = main)
  }
}

# Remove burn-in
nBurn  = max(which(is.na(chain$mean) | is.na(chain2$mean)))
chain  = chain[ -(1:nBurn), ]
chain2 = chain2[-(1:nBurn), ]


# Results
plot_res(chain, chain2, nrow(chain), main = main)
hdi1 = as.numeric(hdi(chain[, 1],  credMass = 0.95))
hdi2 = as.numeric(hdi(chain2[, 1], credMass = 0.95))


2*1.96/sqrt(2e3)
diff(hdi1)
diff(hdi2)

The results, where hdi1 is my "likelihood" and hdi2 is the simple rnorm(n, theta, 1):

> 2*1.96/sqrt(2e3)
[1] 0.08765386
> diff(hdi1)
[1] 1.087125
> diff(hdi2)
[1] 0.07499163

So after lowering the tolerance sufficiently, and at the expense of many more MCMC steps, we can see the expected CrI width for the rnorm model.

enter image description here

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  • $\begingroup$ Not duplicate, but has close relation with stats.stackexchange.com/questions/419916/… $\endgroup$ – user158565 Aug 1 at 18:30
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    $\begingroup$ Generally, when you have an informative prior that is quite wrong, in an informal sense, e.g., Normal(0,1) when the actual value is -3.6, your credible interval in the absence of a lot of data will be pretty poor when looked at from a frequentist perspective. $\endgroup$ – jbowman Aug 1 at 18:46
  • $\begingroup$ @jbowman This is specifically about the case when using a uniform prior or something like N(0, 1e6). $\endgroup$ – Livid Aug 1 at 18:52
  • $\begingroup$ Decades ago, the real Bayesian called the statistician who used non-informative prior as pseudo- (or fake-) Bayesian. $\endgroup$ – user158565 Aug 1 at 19:26
  • $\begingroup$ @user158565 This is offtopic but a uniform prior is just an approximation. If p(H_0) = p(H_1) = p(H_2) = ... = p(H_n) then all the priors can drop out of Bayes' rule making computation easier. It is no more wrong than dropping small terms from a denominator when it makes sense. $\endgroup$ – Livid Aug 1 at 19:29
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In a sequential experimental design, the credible interval can be misleading.

(Disclaimer: I am not arguing it is not reasonable - it is perfectly reasonable in Bayesian reasoning and is not misleading in the perspective of Bayesian point of view.)

For a simple example, let say we have a machine giving us a random sample $X$ from $N(\theta,1)$ with unknown $\theta$. Instead of drawing $n$ i.i.d. samples, we draw samples until $\sqrt{n} \bar{X}_n > k$ for a fixed $k$. That is, the number of samples is a stopping time $N$ defined by $$ N = \inf\left\{n \geq 1 : \sqrt{n}\bar{X}_n > k \right\}. $$

From the law of iterated logarithm, we know $P_{\theta}(N < \infty) = 1$ for any $\theta \in \mathbb{R}$. This type of stopping rule is commonly used in sequential testing/estimations to reduce the number of samples to do inference.

Likelihood principle shows that posterior of $\theta$ is not affected by stopping rule and thus for any reasonable smooth prior $\pi(\theta)$ (e.g., $\theta \sim N(0, 10))$, if we set a large enough $k$, the posterior of $\theta$ is approximately $N(\bar{X}_N, 1/N)$ and thus the credible interval is approximately given as $$ CI_{bayes} :=\left[\bar{X}_N - \frac{1.96}{\sqrt{N}}, \bar{X}_N + \frac{1.96}{\sqrt{N}}\right]. $$ However, from the definition of $N$, we know that this credible interval does not contain $0$ if $k$ is large since $$ 0 < \bar{X}_N - \frac{k}{\sqrt{N}} \ll \bar{X}_N - \frac{1.96}{\sqrt{N}} $$ for $k \gg 0$. Therefore, the frequentist coverage of $CI_{bayes}$ is zero since $$ \inf_{\theta} P_\theta( \theta \in CI_{bayes} ) = 0, $$ and $0$ is attained when $\theta$ is $0$. In contrast, the Bayesian coverage is always approximately equal to $0.95$ since $$ \mathbb{P}( \theta \in CI_{bayes} | X_1, \dots, X_N) \approx 0.95. $$

Take home message: If you are interested in having a frequentist guarantee, you should be careful about using Bayesian inference tools which is always valid for Bayesian guarantees but not always for frequentist ones.

(I learned this example from Larry's awesome lecture. This note contains many interesting discussion about the subtle difference between frequentist and Bayesian frameworks. http://www.stat.cmu.edu/~larry/=stat705/Lecture14.pdf)

EDIT In Livid's ABC, the tolerance value is too large, so even for the standard setting where we sample a fixed number of observations, it does not gives a correct CR. I am not familiar with ABC but if I only changed the tol value to 0.05, we can have a very skewed CR as following

> X = like(theta = 0)
> m = mean(X)
> print(m)
[1] 0.02779672

enter image description here

> as.numeric(hdi(chain[, 1], credMass = 0.95)) [1] -0.01711265 0.14253673

Of course, the chain is not well-stabilized but even if we increase the chain length, we can get similar CR - skewed to positive part.

In fact, I think the rejection rule based on mean difference is not well-suited in this setting since with high probability $\sqrt{N}\bar{X}_N$ is close to $k$ if $0<\theta \ll k$ and close to $-k$ if $-k \ll \theta <0$.

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  • $\begingroup$ "if we set a large enough k, the posterior of θ is approximately N(X_N,1/N)". It seems to me that obviously Pr(X|theta) != Normal(theta, 1). Ie, that is the wrong likelihood for the process that generated your sequence. Also, there is a typo. In the original example you stop sampling when sqrt(n)*abs(mean(x)) > k. $\endgroup$ – Livid Aug 6 at 18:50
  • $\begingroup$ Thanks for comments. Even under the stopping rule, the likelihood is given as $\prod_{i=1}^N \phi(X_i - \theta)$. So it is same as the product of N normal observations though N is now random. This example is still valid for the current stopping rule although the one you pointed out is the original and historical example - There is an interesting history of how frequentist and Bayesian have debated with this stopping rule. You can see, for example, errorstatistics.com/2013/04/06/… $\endgroup$ – JaeHyeok Shin Aug 6 at 19:25
  • $\begingroup$ Please see my edit in the question. I still think your credible interval doesn't make sense because it uses an incorrect likelihood. When using the correct likelihood like in my code we get a reasonable interval. $\endgroup$ – Livid Aug 6 at 20:56
  • $\begingroup$ Thanks for the detailed experiment. In your setting, $k$ is too small to satisfy the inequalities $0 < \bar{X}_N - k / \sqrt{N} \ll \bar{X}_N - 1.96 / \sqrt{N} $. Note that $k$ must be larger than 1.96, and to compansate approximation error, I think $k > 10$ would be safe choice. $\endgroup$ – JaeHyeok Shin Aug 7 at 2:45
  • $\begingroup$ Also, could you re-check whether this ABC computing works for the standard case? I replaced the \code{like} function to make it draw a fixed number of observations (n = 2000) Then, the theoretical length of CR should be approximately $2 \times 1.96 / \sqrt{2000} = 0.0876$ but I always get very wider CR (length is approximately 2). I think the tolerance level is too large. $\endgroup$ – JaeHyeok Shin Aug 7 at 4:18
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Since the credible interval is formed from the posterior distribution, based on a stipulated prior distribution, you can easily construct a very bad credible interval by using a prior distribution that is heavily concentrated on highly implausible parameter values. You can make a credible interval that does not "make sense" by using a prior distribution that is entirely concentrated on impossible parameter values.

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  • 1
    $\begingroup$ Or better yet, a credible constructed by a prior that disagrees with your prior (even though it is someone else's prior) has good odds with being preposterous to you. This is not uncommon in science; I've had researchers say they don't want to include expert's opinion, because in their observations, the experts have always been strongly overconfident. $\endgroup$ – Cliff AB Aug 7 at 3:43
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    $\begingroup$ This is specifically about uniform, or "flat", priors. $\endgroup$ – Livid Aug 7 at 8:02
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    $\begingroup$ @Livid: You should definitely include that you are talking about flat priors in your question. That completely changes everything. $\endgroup$ – Cliff AB Aug 7 at 16:39
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    $\begingroup$ @CliffAB It is in the first two sentences, but I will clarify, thanks. $\endgroup$ – Livid Aug 7 at 16:58
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If we are using a flat prior, this is simply a game where we try to come up with a flat prior in a reparameterization that doesn't make sense.

For example, suppose we want to make inference on a probability. If we put a flat prior on the log odds of the probability, our 95% credible interval for the actual probability is the two points $\{0,1\}$ before we even see the data! If we get a single positive data point and construct a 95% credible interval, it is now the single point $\{1\}$.

This is why many Bayesians object to flat priors.

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  • $\begingroup$ I explained my motivation pretty clearly. I want something like the examples where confidence intervals include impossible values, but where the credible interval behaves fine. If your example hinges on doing something nonsensical, like eg choosing the wrong likelihood, then why would it be of interest to anyone? $\endgroup$ – Livid Aug 7 at 17:27
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    $\begingroup$ @Livid: the likelihood function is perfectly reasonable. The flat prior on the log-odds is not. And this is the entirety of the argument that Bayesian use to say you shouldn't use flat priors; they can actually be extremely informative and often not as the user intended! $\endgroup$ – Cliff AB Aug 7 at 17:39
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    $\begingroup$ Here's Andrew Gelman discussing some of the issues of flat priors. $\endgroup$ – Cliff AB Aug 7 at 17:40
  • $\begingroup$ "The flat prior on the log-odds is not." I meant that putting a flat prior on log-transformed odds seems to be nonsense to you, like using the wrong likelihood. Sorry, but I am not familiar with this example. What is this model supposed to do exactly? $\endgroup$ – Livid Aug 7 at 17:49
  • $\begingroup$ @Livid: it may seem unusual, but it's really not! For example, logistic regression typically considers all the parameters on the log-odds scale. If you had dummy variables for all your groups and used flat priors on your regression parameters, you would run into exactly this issue. $\endgroup$ – Cliff AB Aug 7 at 18:14

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