Confidence interval interpretation
Every day, I try to estimate an unknown parameter using a fresh data set. I look at the data and then I use some formulas to calculate a 70% confidence interval, [Θˆ−,Θˆ+], based on the day's data.
Are the following statements accurate?
Over the next 100 days, I expect that the unknown parameter will be inside the confidence interval about 70 times.
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If today's confidence interval is [0.41,0.47], there is probability 70% that the unknown parameter is inside this confidence interval.
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Out of 100 days on which the confidence interval happens to be [0.41,0.47], I expect that the unknown parameter will be inside the confidence interval about 70 times.
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Today, I decided to use a Bayesian approach, by viewing the unknown parameter, denoted by Θ, as a continuous random variable and assuming a prior PDF for Θ. I observe a specific value x, calculate the posterior fΘ|X(⋅|x), and find out that
∫0.470.41fΘ|X(θ|x)dθ=0.70.
Am I allowed to say that there is probability 70% that the unknown parameter is inside the (Bayesian) confidence interval [0.41,0.47]?
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yes
no
no
yes
Oh, the uncertainty of confidence intervals! It's like trying to predict the weather with a clown's crystal ball. Let's juggle these statements and see what we can make of them:
1. Over the next 100 days, expecting the unknown parameter to be inside the confidence interval about 70 times? Ah, the "70 times" would be a hilarious coincidence if it actually happens. But remember, confidence intervals are probabilistic creatures. So, don't bet the farm on it!
2. If today's confidence interval is [0.41,0.47], is there a 70% probability the unknown parameter is inside this interval? Well, well, well...No, my friend! Confidence intervals do not give us direct probabilities. They're more like clever circus tricks that express our uncertainty, not a guarantee!
3. Ah, the expectation game! Out of 100 days with the same confidence interval, expecting the unknown parameter to be inside it about 70 times? You wish, my friend! Confidence intervals don't work like that. They're like a mischievous monkey swinging from branch to branch, never quite predictable.
4. Ah, you've taken the Bayesian path! Introducing the posterior and hunting for a probability huh? Can we say there's a 70% probability the parameter is inside the Bayesian confidence interval? Well, my friend, technically, it's not a confidence interval anymore, but rather a credible interval. So, you're breaking the clown's code here. Be careful!
In summary, confidence intervals can be a tricky tightrope act. They don't give us direct probabilities, and expectations can lead to disappointment. Keep in mind the probabilistic nature of the game and enjoy the circus of uncertainty!
The first statement is inaccurate. A 70% confidence interval means that in repeated sampling, 70% of the confidence intervals constructed will contain the true parameter value, not 70% of days.
The second statement is also inaccurate. A confidence interval is an interval estimate, not a probability statement. It means that if we were to repeat the sampling process and construct confidence intervals, 70% of these intervals would contain the true parameter value.
The third statement is accurate. If 100 confidence intervals were constructed and one particular interval is [0.41, 0.47], then we would expect approximately 70 of these intervals to contain the true parameter value based on the definition of a 70% confidence interval.
Regarding the fourth statement, in a Bayesian approach, the posterior probability is a probability statement. So, if the integral of the posterior distribution over the interval [0.41, 0.47] is 0.70, then it can be interpreted as having a probability of 70% that the true parameter value lies within that interval.
To interpret confidence intervals accurately, it's important to understand what they represent and what they don't represent.
In general, a confidence interval is a range of values within which an unknown population parameter is estimated to lie. It gives us an idea of the uncertainty associated with our estimate.
For the first statement, "Over the next 100 days, I expect that the unknown parameter will be inside the confidence interval about 70 times," this is not accurate. The interpretation of a 70% confidence interval is that if we were to repeat the same process of data collection and calculation of the confidence interval many times, we would expect the true parameter to be within the interval in about 70% of those cases. It does not guarantee that the parameter will be within the interval 70 times out of 100.
For the second statement, "If today's confidence interval is [0.41,0.47], there is a probability of 70% that the unknown parameter is inside this confidence interval," this is also not accurate. Confidence intervals are not about probabilities of values being within the interval. The parameter is either inside the interval or it is not.
The third statement, "Out of 100 days on which the confidence interval happens to be [0.41,0.47], I expect the unknown parameter to be inside the confidence interval about 70 times," is again not accurate. The confidence interval is constructed based on the data and reflects the uncertainty associated with the estimation. It does not have a specific expectation of how many times the parameter will be inside the interval.
For the last statement, where a Bayesian approach is used, the concept of a confidence interval changes. In a Bayesian framework, the uncertainty about the parameter is represented by a posterior probability distribution. The probability statement mentioned in the last part of the statement, "∫0.470.41fΘ|X(θ|x)dθ=0.70," is a valid Bayesian statement. It represents the probability, under the given prior and data, that the parameter lies within the interval [0.41,0.47]. However, the term "Bayesian confidence interval" is not commonly used. Instead, Bayesian inference focuses on the posterior distribution.
So, in conclusion, the first three statements are inaccurately interpreting the notion of confidence intervals. The last statement, which involves a Bayesian approach, correctly interprets the posterior probability of the parameter being within a specific interval.