Dirichlet Distribution | Intuition & Intro | w\ example in TensorFlow Probability

The parameter to the Categorical is a vector of parameters. Can we put a distribution on it? Yes, we can. That's the Dirichlet. Here are the notes: https://raw.githubusercontent.com/Ceyron/machine-learning-and-simulation/main/english/essential_pmf_pdf/dirichlet_intro.pdf

The Parameter vector to the Categorical is of the dimension equal to the number of states of the Categorical. For example, we model the weather as the three states: Cloudy, Rainy or Sunny then we need a parameter (a probability) for each of the states.

There are two requirements on this probability vector: (1) all entries must be chosen from the interval [0, 1] (since they are probabilities), (2) the vector's components have to sum up to one. In this video, we will see that this implies the that the D-dimensional parameter vector is distributed over a (D-1)-dimensional simplex in D dimensions.

The Dirichlet describes a probability density distribution over this simplex. It is parameterized by an alpha-vector with also D-components which we can use to move the probability mass around over the simplex

Here is the website I showed in the video:
https://chart-studio.plotly.com/~david_avakian/14.embed

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Timestamps:
00:00 Introduction
00:33 Restrictions on the Parameter Vector
02:00 Visualizing 2-State Parameter Vector
05:56 Connection to the Beta Distribution
07:03 Visualizing 3-State Parameter Vector
09:25 General D-State Parameter Vector
10:37 Probability Density Function
11:56 Parameters of the Dirichlet
12:24 Plot: Exploring alpha values
15:58 TFP: Creating the Dirichlet Distribution
16:49 TFP: Sampling the Dirichlet
17:16 TFP: Querying the pdf
17:55 TF: Calculating Multivariate Beta Function
18:47 Outro

Видео Dirichlet Distribution | Intuition & Intro | w\ example in TensorFlow Probability канала Machine Learning & Simulation