1

Recently, I came across the following image of a "probability distribution" defined over a "simplex":

enter image description here

Question: In the above picture, what exactly is the "simplex"?

I tried reading about the official definition of a "simplex":

enter image description here

However, I can't seem to understand the relationship between the "simplex" in which the probability distribution is defined compared to the official definition of the "simplex".

Can someone please help me understand what exactly is a "simplex" in this context? Does this "simplex" have any relationship to the "simplex method" (https://en.wikipedia.org/wiki/Simplex_algorithm) in optimization and linear programming?

enter image description here

References:

stats_noob
  • 5,882
  • 1
  • 21
  • 42
  • 3
    The "triangular surface" referred to in Fig. 2.8 is a *projection* of the simplex in $\mathbb{R}^{n+1}$ to $\mathbb{R}^n.$ One convenient such projection is described in my answer at https://stats.stackexchange.com/a/259223/919. The projection in Fig 2.8 simply ignores the $n+1^\text{st}$ coordinate. – whuber Nov 13 '21 at 21:30
  • @ Whuber : thank you for your reply! I am still trying to understand what exactly a "simplex" is ... but in the first picture of my post, was it really necessary to define this probability distribution over a "simplex"? Could they not have just defined it over a domain of x-values? (i.e. the support) – stats_noob Nov 14 '21 at 00:32
  • No, because of the constraint that the x-values add to 1. – jbowman Nov 14 '21 at 03:03
  • 1
    One of the **uses** of a simplex goes something like this: if you have a coordinate point in $D$ dimensions, a simplex is a sex of $D+1$ points which contain the coordinate point. This was used, for example, by Sugihara in "simplex projection" based on Takens Theorem, where in a $D$ dimensional state space, a $D$-dimensional coordinate representing a position on a manifold projection of the state space can be "contained" within the smallest $D+1$ point simplex from the history of points in the manifold, and their centroid represents the best estimate of the next point in the state space. – Alexis Nov 14 '21 at 04:39
  • See for example, https://www.youtube.com/watch?v=by5l-ljo-SI – Alexis Nov 14 '21 at 04:40
  • 2
    Cross-posted: https://math.stackexchange.com/questions/4305547/how-the-geometric-definition-of-simplex-relates-to-the-probability-distributio – Peter O. Nov 14 '21 at 05:59

0 Answers0