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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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