# Uncertainty Wednesday: Beliefs (Cont’d)

By [Continuations](https://continuations.com) · 2018-05-02

uncertainty wednesday, beliefs

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Last [Uncertainty Wednesday](https://continuations.com/tagged/uncertainty-wednesday), I introduced the idea of [beliefs](https://continuations.com/post/173307469685/uncertainty-wednesday-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)

![image](https://storage.googleapis.com/papyrus_images/9630fd27bb4ba2ef30bc98dea247f0fd.png)

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). 

The chart is the [probability density function](https://en.wikipedia.org/wiki/Probability_density_function) (pdf) for the [beta distribution](https://en.wikipedia.org/wiki/Beta_distribution) with parameters alpha = beta = 2. You might wonder why the graph goes to values above 1, which would seem to suggest probabilities > 1, but the whole area under the curve is exactly 1. Probabilities are derived from the pdf as small slivers around a value of p (the horizontal axis). For instance between p = 0.45 and p = 0.55 if you imagine vertical lines you get an area of approximately 0.1 \* 1.5 = 0.15. So with this belief we are saying that the coin has about a 15% chance of being pretty fair.

Another way to show this belief is via the following cumulative distribution function (cdf):

![image](https://storage.googleapis.com/papyrus_images/1a539ad1d5d2be826116b9b78733ed65.png)

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:

![](https://storage.googleapis.com/papyrus_images/dc27927f55128b8651927af802caef70.png)

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](https://en.wikipedia.org/wiki/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.

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*Originally published on [Continuations](https://continuations.com/uncertainty-wednesday-beliefs-contd)*