Inverted Dirichlet distribution
Okay kiddo, an inverted Dirichlet distribution is a way of describing the probability of picking different options when you have to pick from a bunch of options.
Let's imagine that we have a bunch of different colored balls in a bag. We can pick one ball out of the bag at a time, and each time we pick a ball, the chance of us picking a particular color is different. The inverted Dirichlet distribution tells us what those chances are.
The inverted Dirichlet distribution works by giving each color in the bag a "weight". This weight tells us how likely we are to pick a ball of that color. So, a ball with a weight of 1 is just as likely to be picked as a ball with a weight of 2, but a ball with a weight of 3 is three times more likely to be picked than a ball with a weight of 1.
Now, when we say "inverted" Dirichlet distribution, we're talking about the weights themselves. Normally, when we use a Dirichlet distribution, the weights have to add up to 1. This makes sense, because if we're picking a ball out of a bag, we have to pick one of the balls. But an inverted Dirichlet distribution is different - the weights have to add up to a number bigger than 1!
Why would we want to do this? It turns out that there are lots of situations where we want to describe probabilities that are "biased" towards some outcomes. For example, if we're trying to simulate a sports game, we might want to make it more likely for the home team to win. Or if we're trying to predict which movie will win an Oscar, we might want to give more weight to the movies that have already won lots of other awards.
So, an inverted Dirichlet distribution is a way of describing these biased probabilities. We start by giving each outcome (like picking a ball of a certain color) a weight, and then we use that weight to calculate the probability of each outcome. It's sort of like having a "magic bag" where some balls are more likely to be picked than others!
Does that make sense, kiddo?
Let's imagine that we have a bunch of different colored balls in a bag. We can pick one ball out of the bag at a time, and each time we pick a ball, the chance of us picking a particular color is different. The inverted Dirichlet distribution tells us what those chances are.
The inverted Dirichlet distribution works by giving each color in the bag a "weight". This weight tells us how likely we are to pick a ball of that color. So, a ball with a weight of 1 is just as likely to be picked as a ball with a weight of 2, but a ball with a weight of 3 is three times more likely to be picked than a ball with a weight of 1.
Now, when we say "inverted" Dirichlet distribution, we're talking about the weights themselves. Normally, when we use a Dirichlet distribution, the weights have to add up to 1. This makes sense, because if we're picking a ball out of a bag, we have to pick one of the balls. But an inverted Dirichlet distribution is different - the weights have to add up to a number bigger than 1!
Why would we want to do this? It turns out that there are lots of situations where we want to describe probabilities that are "biased" towards some outcomes. For example, if we're trying to simulate a sports game, we might want to make it more likely for the home team to win. Or if we're trying to predict which movie will win an Oscar, we might want to give more weight to the movies that have already won lots of other awards.
So, an inverted Dirichlet distribution is a way of describing these biased probabilities. We start by giving each outcome (like picking a ball of a certain color) a weight, and then we use that weight to calculate the probability of each outcome. It's sort of like having a "magic bag" where some balls are more likely to be picked than others!
Does that make sense, kiddo?