### (left) Uniform; (right) Normal / Gaussian  ### What we interested in knowing is actually p(x|y). This is called the posterior distribution of the unknown variable x given our known variable y. Luckily, there's a pretty famous formula to help us calculate this: it's called Bayes Rule. ### Okay, let's look at an example in a spreadsheet. Imagine you have one cell describing your prior for a unknown variable x. The cell contains the formula GAUSSIAN(1, 5), describing a normal distribution centered at 1 with a standard deviation of 5. ### Now let's use Invrea Scenarios to generate 100,000 scenarios and see what this prior distribution looks like. We should expect a peak around 1. ### Invrea Scenarios Excel Ribbon ### Indeed, it looks as expected, and we have an expected value for x at 0.997. Now let's add our data, y. This is a known variable, and let's set it to something far away from our prior, like 9. What this is suggesting is that our prior was a bad guess. Let's see if Bayes' rule can help us fix it. In Invrea Scenarios, we use the following formula to define y: ACTUAL(9, "gaussian", x, std), where 9 is the known data point, and the rest of the parameters define the conditional distribution for y given x. Let's rerun 100,000 scenarios and see what the posterior distribution looks like after considering this new data point.  ### Invrea Scenarios Excel Ribbon 