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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Sure, let's give these statements a humorous twist!
1. Over the next 100 days, I expect that the unknown parameter will be inside the confidence interval about 70 times.
Well, you may be confident in your 70% confidence interval, but remember, life is unpredictable. It's like trying to predict how many times a dog will bark in the next 100 days. Sometimes it's quiet, and sometimes it's a party of barks. So, don't place all your bets on that 70!
2. 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.
Ah, the mysterious world of statistics, where confidence intervals are like doors to the unknown. But remember, probability and confidence are like frenemies. You can't say with certainty that the unknown parameter is inside the interval, but hey, with 70% probability, at least there's a good chance! It's like saying, "There's a 70% probability that my socks will match today." Fingers crossed!
3. 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.
Ah, expectations can be tricky, my friend. Just because the confidence interval stays the same for 100 days, doesn't mean the unknown parameter will play by the rules. It's like expecting your favorite ice cream shop to have 70 flavors every time you visit, but sometimes they only have 30. Keep an open mind, for the unknown parameter is a mischievous creature!
4. Am I allowed to say that there is a probability of 70% that the unknown parameter is inside the (Bayesian) confidence interval [0.41,0.47]?
Ah, the thrill of the Bayesian approach! While you may have calculated the probability, saying with certainty that the unknown parameter is inside the confidence interval is a bit like saying, "There's a 70% chance I'll find buried treasure if I dig in my neighbor's flower bed." It's always an adventure, my friend, but remember to embrace the uncertainty!
No, the statements are not accurate. Confidence intervals are a frequentist concept and should not be interpreted as probabilities.
The interpretation of a confidence interval is that if you were to repeat the data collection process many times and calculate a confidence interval each time, approximately 70% of those intervals would contain the true unknown parameter. It does not mean that the unknown parameter will be inside the confidence interval about 70 times out of 100.
Similarly, with a Bayesian approach, the posterior distribution represents our updated knowledge about the unknown parameter after observing the data, but it does not directly give us probabilities of the parameter being inside a specific interval. Bayesian credible intervals could be used to express uncertainty about the parameter, but they are not the same as frequentist confidence intervals.
To answer your questions, let me explain the interpretation of confidence intervals and how they differ from Bayesian intervals.
A confidence interval provides a range of values within which we can be reasonably confident that the true population parameter lies. In your case, you calculate a 70% confidence interval based on each day's data.
1. The statement "Over the next 100 days, I expect that the unknown parameter will be inside the confidence interval about 70 times" is not accurate. A 70% confidence interval does not guarantee that the parameter will be inside the interval 70% of the time. Instead, it means that if you repeatedly sample from the population and construct 100 confidence intervals, approximately 70 of them will contain the true parameter.
2. Similarly, the 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" is not accurate. Confidence intervals are not based on probabilities but on the concept of repeated sampling. The true parameter is either in the interval or not; it does not have a probability of being in the interval.
3. The statement "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" is also not accurate. The confidence intervals are constructed based on the data from each day independently. Therefore, there is no guarantee that the true parameter will be inside the interval a specific number of times.
Now, let's talk about Bayesian intervals. In Bayesian statistics, intervals represent probabilities and can be interpreted differently. In your case, you used a Bayesian approach, and you have calculated the posterior distribution for the unknown parameter given the observed data.
4. In the Bayesian framework, when you integrate the posterior distribution over a specific interval, you can say that there is a probability associated with the interval. So, in your case, when you integrate the posterior from 0.41 to 0.47 and get a value of 0.70, you can say that there is a 70% probability that the unknown parameter lies within that interval.
To summarize, confidence intervals and Bayesian intervals have different interpretations. Confidence intervals are based on repeated sampling and do not provide probabilities for individual intervals. Bayesian intervals, on the other hand, can be interpreted as probabilities based on the posterior distribution.