Now that we've had some exposure to Bayesian approaches, let's pause and think about how these compare to frequentist approaches. Consider these three statements. The probability of flipping a coin and getting heads is one-half. The probability of rolling snake eyes, that is, two 1s on two dice, is 1/36. The probability of Apple's stock price going up today is 0.75. What exactly do these statements mean? How you interpret these statements depends on your definition of probability. One definition of probability of an event is its relative frequency in a large number of trials. For example, if you can repeat flipping a coin indefinitely and count how many heads you get and divide that number by the number of flips, the value you obtain should be 0.5. In other words the probability of event E is defined as the proportion of times the event occurs and n trials when n goes to infinity. This is the frequentist definition of probability, suppose now that you're indifferent between winning a dollar if event E occurs or winning a dollar if you draw a blue chip from a box with 1,000 x p blue chips and 1,000 x (1-p) white chips. This means that you're equating the probability of events E, that's P(E), to the probability of drawing a blue chip from this box. In other words P(E) = p. This definition of probability, based on your degree of belief, is the Bayesian definition. So, what are the implications of these two different definitions? In earlier courses in this specialization, we talked about frequentists methods of inference, for example, a confidence interval. When defining the confidence level we were very careful to describe it as the proportion of random samples of size n from the same population that produced confidence intervals that contain the true population parameter. We emphasized that an interpretation of the confidence level as the probability that a given interval containing the true parameter is incorrect. For example, based on a 2015 Pew Research poll on 1500 adults, we created the following confidence interval. We are 95% confident that 60% to 64% of Americans think the federal government does not do enough for middle class people. What does 95% confident mean in this statement? The correct answer is that 95% of random samples of 1,500 adult Americans will produce confidence intervals for the proportion of Americans who think the federal government does not do enough for a middle class people. However some common misconceptions about the confidence level interpret this value as there's a 95% chance that this confidence interval includes the true population proportion. Or the true population proportion is in this interval 95% of the time. The frequentist definition of probability allows to define a probability for the confidence interval procedure but not for specific fixed sample. And the case of a specific fixed sample, when the data do not change, we will either always capture the true parameter or never capture it. In other words, for given confidence interval the true parameter is either in it or not. This is the same as saying that the probability that a given confidence interval captures the true parameter, is either zero or one. The only problem is that we can't know whether the probability that this given interval captures the true parameter is zero or one. The Bayesian definition is a bit more flexible. Since it's a measure of belief it allows us to describe the unknown true parameter not as a fixed value but with a probability distribution. This will let us construct something like a confidence interval, except we will be able to make probabilistic statements about the parameter falling within that range. For example, we could say something like the posterior distribution yields a 95% credible interval of 60% to 64% for the proportion of Americans who think the federal government does not do enough for middle class people. These are called credible intervals and we will introduce them later in the course.