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Last Uncertainty Wednesday, I introduced the idea of beliefs. Today we will make this idea more precise. We started with an extreme belief, the one that a coin is so biased that we will only observe “heads” (H). More realistically one might belief that a coin is fair, but has some possibility of being slightly biased in either direction (e.g., more likely to observer H or more likely to observe T).
So how do we formalize this? A belief is simply a probability distribution. For instance the one we just described might be modeled as follows where the horizontal axis shows the values of p from 0 to 1 (where p is the probability of observing Heads)
This is a distribution with a mean at p = 0.5 where the probability decreases to 0 on either extreme (meaning at p = 0 and at p = 1).
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Last Uncertainty Wednesday, I introduced the idea of beliefs. Today we will make this idea more precise. We started with an extreme belief, the one that a coin is so biased that we will only observe “heads” (H). More realistically one might belief that a coin is fair, but has some possibility of being slightly biased in either direction (e.g., more likely to observer H or more likely to observe T).
So how do we formalize this? A belief is simply a probability distribution. For instance the one we just described might be modeled as follows where the horizontal axis shows the values of p from 0 to 1 (where p is the probability of observing Heads)
This is a distribution with a mean at p = 0.5 where the probability decreases to 0 on either extreme (meaning at p = 0 and at p = 1).
Another way to show this belief is via the following cumulative distribution function (cdf):
For each value of p along the horizontal axis, the cdf shows the integral of the pdf from 0 to that value of p, where p = 0.5 represents a fair coin. Looking at it this way we can see how much probability we attribute to the coin being less or more biased than a specific value of p.
Contrast this with a belief that the coin is precisely fair, which would have a cumulative distribution function that looks like this instead:
Here the entire probability is concentrated on p = 0.5! We believe with 100% certainty that the coin is exactly fair. We attribute no probability to it being biased in either direction. The bottom horizontal line ends at 0.5 with a blank circle ○ and the top horizontal line starts at 0.5 with a solid circle ●, indicating that the value of the function jumps from 0 to 1 at 0.5. So why not draw a probability density function? The reason is that technically we would have to show something different, namely a probability mass function.
At a later point we will see just how extreme such a belief is, but even just looking at a discontinuous function should provide an inkling of that.
Another way to show this belief is via the following cumulative distribution function (cdf):
For each value of p along the horizontal axis, the cdf shows the integral of the pdf from 0 to that value of p, where p = 0.5 represents a fair coin. Looking at it this way we can see how much probability we attribute to the coin being less or more biased than a specific value of p.
Contrast this with a belief that the coin is precisely fair, which would have a cumulative distribution function that looks like this instead:
Here the entire probability is concentrated on p = 0.5! We believe with 100% certainty that the coin is exactly fair. We attribute no probability to it being biased in either direction. The bottom horizontal line ends at 0.5 with a blank circle ○ and the top horizontal line starts at 0.5 with a solid circle ●, indicating that the value of the function jumps from 0 to 1 at 0.5. So why not draw a probability density function? The reason is that technically we would have to show something different, namely a probability mass function.
At a later point we will see just how extreme such a belief is, but even just looking at a discontinuous function should provide an inkling of that.
Albert Wenger
Albert Wenger
>300 subscribers
>300 subscribers
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