# Confidence Intervals They key confusion about confidence intervals really just comes down to what you are treating as ***fixed*** and what you are allowing to vary:  The **frequentist** approach is to view the mean (our parameter, $\theta$) as fixed, with the sample being random. Which leads to their interpretation: > 95% of confidence intervals constructed would contain the true population mean. An interval constructed like the above is what is called a confidence interval. Since, the true value is unknown but fixed, the true value is either in the interval or outside the interval. The confidence interval then is a statement about the likelihood that the interval we obtain actually has the true parameter value. Thus, the probability statement is about the interval (i.e., the chances that interval which has the true value or not) rather than about the location of the true parameter value. While on the other hand, the **bayesian** approach is to have $\theta$ be a rv, with a distribution. The sample (the data) is fixed. Hence, we now are say: > There is a 95% chance that the interval contains the true population mean. Or, equivalently: > There is a 95% chance that the true population mean (a rv) is in the interval. --- Date: 20220210 Links to: [Frequentist vs Bayesian](Frequentist%20vs%20Bayesian.md) Tags: References: * [My blog, confidence intervals](https://www.nathanieldake.com/Mathematics/04-Statistics-03-statistical-inference.html#3.2-Confidence-Intervals) * [My blog, bayesian approach](https://www.nathanieldake.com/Machine_Learning/08-Bayesian_Machine_Learning-02-Bayesian-AB-Testing.html#2.2-UCB1) * [What are confidence intervals? Actually. - YouTube](https://youtu.be/EJe3jiZNwUU?t=1362) *