Mixing
What is the difference between a unit simplex and a probability simplex?
$begingroup$
The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, ldots,e_ninmathbf R^n$. It can be expressed as the set of vectors that satisfy
$$xsucccurlyeq0,quadmathbf 1^mathrm T xle1.$$
The probability simplex is the $(n−1)$-dimensional simplex determined by the unit vectors $e_1,ldots ,e_ninmathbf R^n$. It is the set of vectors that satisfy
$$xsucccurlyeq 0,quad mathbf 1^mathrm T x=1.$$
I know that a simplex is the set of all convex combinations of some vectors. I can imagine that in two dimensions, the probability simplex is a right triangle with legs $mathbf e_1$ and $mathbf e_2$. In three dimensions, it is a right tetrahedron with legs $mathbf e_1$, $mathbf e_2$ and $mathbf e_3$. But what does the unit simplex look like? What difference can the zero vector make?
simplex convex-geometry
edited Jan 17 at 2:34
El borito
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
$endgroup$
add a comment |
$begingroup$
The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, ldots,e_ninmathbf R^n$. It can be expressed as the set of vectors that satisfy
$$xsucccurlyeq0,quadmathbf 1^mathrm T xle1.$$
The probability simplex is the $(n−1)$-dimensional simplex determined by the unit vectors $e_1,ldots ,e_ninmathbf R^n$. It is the set of vectors that satisfy
$$xsucccurlyeq 0,quad mathbf 1^mathrm T x=1.$$
I know that a simplex is the set of all convex combinations of some vectors. I can imagine that in two dimensions, the probability simplex is a right triangle with legs $mathbf e_1$ and $mathbf e_2$. In three dimensions, it is a right tetrahedron with legs $mathbf e_1$, $mathbf e_2$ and $mathbf e_3$. But what does the unit simplex look like? What difference can the zero vector make?
simplex convex-geometry
edited Jan 17 at 2:34
El borito
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
$endgroup$
$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12
add a comment |
$begingroup$
The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, ldots,e_ninmathbf R^n$. It can be expressed as the set of vectors that satisfy
$$xsucccurlyeq0,quadmathbf 1^mathrm T xle1.$$
The probability simplex is the $(n−1)$-dimensional simplex determined by the unit vectors $e_1,ldots ,e_ninmathbf R^n$. It is the set of vectors that satisfy
$$xsucccurlyeq 0,quad mathbf 1^mathrm T x=1.$$
I know that a simplex is the set of all convex combinations of some vectors. I can imagine that in two dimensions, the probability simplex is a right triangle with legs $mathbf e_1$ and $mathbf e_2$. In three dimensions, it is a right tetrahedron with legs $mathbf e_1$, $mathbf e_2$ and $mathbf e_3$. But what does the unit simplex look like? What difference can the zero vector make?
simplex convex-geometry
edited Jan 17 at 2:34
El borito
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
$endgroup$
The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, ldots,e_ninmathbf R^n$. It can be expressed as the set of vectors that satisfy
$$xsucccurlyeq0,quadmathbf 1^mathrm T xle1.$$
The probability simplex is the $(n−1)$-dimensional simplex determined by the unit vectors $e_1,ldots ,e_ninmathbf R^n$. It is the set of vectors that satisfy
$$xsucccurlyeq 0,quad mathbf 1^mathrm T x=1.$$
I know that a simplex is the set of all convex combinations of some vectors. I can imagine that in two dimensions, the probability simplex is a right triangle with legs $mathbf e_1$ and $mathbf e_2$. In three dimensions, it is a right tetrahedron with legs $mathbf e_1$, $mathbf e_2$ and $mathbf e_3$. But what does the unit simplex look like? What difference can the zero vector make?
simplex convex-geometry
simplex convex-geometry
edited Jan 17 at 2:34
El borito
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
edited Jan 17 at 2:34
El borito
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
edited Jan 17 at 2:34
El borito
666216
edited Jan 17 at 2:34
El borito
666216
edited Jan 17 at 2:34
El borito
666216
666216
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
asked Jan 17 at 2:29
W. ZhuW. Zhu
685316
685316
$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12
add a comment |
$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12
$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12
$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
From your definition, the probability simplex is the subset of the unit simplex in which the sum of the elements of the vector is exactly one, i.e., $sum_{i=1}^n x_i = 1$.
In two dimensions, the unit simplex is the triangle formed by coordinates (0,0), (0,1) and (1,0), whereas the probability simplex is the line joining (1,0) and (0,1).
Note that the probability simplex has one dimension less than the unit simplex. This is precisely because the probability simplex is constrained by $sum_{i=1}^n x_i = 1$, so you lose one degree of freedom. In the two dimensional case, when you choose a value for $x_1$, $x_2$ is immediately pinned down ($x_2 = 1 - x_1$) in the probability simplex. For the unit simplex, in contrast, $x_2 leq 1 - x_1$ is not pinned down by $x_1$.
edited Jan 18 at 11:17
W. Zhu
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
$endgroup$
add a comment |
$begingroup$
The vectors ${x_i}_{1le ile n+1}$ comprise a regular simplex. Because all vertices are contained within the hyperplane with the equation
$$sum_{1le ile n+1}a_i x_i=1$$
it clearly is just $n$D. Its dihedral angle measured across the margins equally is given by $$arccos(1/n)$$
All edges of that regular simplex obviously have size $sqrt 2$.
When adjoining to the origin, then you'll still have a simplex, then sure being $(n+1)$D. But the slope of the lacing facets here is not so steep as would for the regular one (of according dimension). In fact the lacing margins would clearly show up dihedral angles of $90°$ each. - The size of the base edges here still is $sqrt 2$, while the size of the lacing edges, i.e. those connecting to the origin, clearly is just $1$.
--- rk
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
$endgroup$
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
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$begingroup$
From your definition, the probability simplex is the subset of the unit simplex in which the sum of the elements of the vector is exactly one, i.e., $sum_{i=1}^n x_i = 1$.
In two dimensions, the unit simplex is the triangle formed by coordinates (0,0), (0,1) and (1,0), whereas the probability simplex is the line joining (1,0) and (0,1).
Note that the probability simplex has one dimension less than the unit simplex. This is precisely because the probability simplex is constrained by $sum_{i=1}^n x_i = 1$, so you lose one degree of freedom. In the two dimensional case, when you choose a value for $x_1$, $x_2$ is immediately pinned down ($x_2 = 1 - x_1$) in the probability simplex. For the unit simplex, in contrast, $x_2 leq 1 - x_1$ is not pinned down by $x_1$.
edited Jan 18 at 11:17
W. Zhu
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
$endgroup$
add a comment |
$begingroup$
From your definition, the probability simplex is the subset of the unit simplex in which the sum of the elements of the vector is exactly one, i.e., $sum_{i=1}^n x_i = 1$.
In two dimensions, the unit simplex is the triangle formed by coordinates (0,0), (0,1) and (1,0), whereas the probability simplex is the line joining (1,0) and (0,1).
Note that the probability simplex has one dimension less than the unit simplex. This is precisely because the probability simplex is constrained by $sum_{i=1}^n x_i = 1$, so you lose one degree of freedom. In the two dimensional case, when you choose a value for $x_1$, $x_2$ is immediately pinned down ($x_2 = 1 - x_1$) in the probability simplex. For the unit simplex, in contrast, $x_2 leq 1 - x_1$ is not pinned down by $x_1$.
edited Jan 18 at 11:17
W. Zhu
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
$endgroup$
add a comment |
$begingroup$
From your definition, the probability simplex is the subset of the unit simplex in which the sum of the elements of the vector is exactly one, i.e., $sum_{i=1}^n x_i = 1$.
In two dimensions, the unit simplex is the triangle formed by coordinates (0,0), (0,1) and (1,0), whereas the probability simplex is the line joining (1,0) and (0,1).
Note that the probability simplex has one dimension less than the unit simplex. This is precisely because the probability simplex is constrained by $sum_{i=1}^n x_i = 1$, so you lose one degree of freedom. In the two dimensional case, when you choose a value for $x_1$, $x_2$ is immediately pinned down ($x_2 = 1 - x_1$) in the probability simplex. For the unit simplex, in contrast, $x_2 leq 1 - x_1$ is not pinned down by $x_1$.
edited Jan 18 at 11:17
W. Zhu
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
$endgroup$
From your definition, the probability simplex is the subset of the unit simplex in which the sum of the elements of the vector is exactly one, i.e., $sum_{i=1}^n x_i = 1$.
In two dimensions, the unit simplex is the triangle formed by coordinates (0,0), (0,1) and (1,0), whereas the probability simplex is the line joining (1,0) and (0,1).
Note that the probability simplex has one dimension less than the unit simplex. This is precisely because the probability simplex is constrained by $sum_{i=1}^n x_i = 1$, so you lose one degree of freedom. In the two dimensional case, when you choose a value for $x_1$, $x_2$ is immediately pinned down ($x_2 = 1 - x_1$) in the probability simplex. For the unit simplex, in contrast, $x_2 leq 1 - x_1$ is not pinned down by $x_1$.
edited Jan 18 at 11:17
W. Zhu
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
edited Jan 18 at 11:17
W. Zhu
685316
edited Jan 18 at 11:17
W. Zhu
685316
edited Jan 18 at 11:17
W. Zhu
685316
685316
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
answered Jan 17 at 2:47
Francisco AndrésFrancisco Andrés
556
556
add a comment |
add a comment |
$begingroup$
The vectors ${x_i}_{1le ile n+1}$ comprise a regular simplex. Because all vertices are contained within the hyperplane with the equation
$$sum_{1le ile n+1}a_i x_i=1$$
it clearly is just $n$D. Its dihedral angle measured across the margins equally is given by $$arccos(1/n)$$
All edges of that regular simplex obviously have size $sqrt 2$.
When adjoining to the origin, then you'll still have a simplex, then sure being $(n+1)$D. But the slope of the lacing facets here is not so steep as would for the regular one (of according dimension). In fact the lacing margins would clearly show up dihedral angles of $90°$ each. - The size of the base edges here still is $sqrt 2$, while the size of the lacing edges, i.e. those connecting to the origin, clearly is just $1$.
--- rk
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
$endgroup$
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
add a comment |
$begingroup$
The vectors ${x_i}_{1le ile n+1}$ comprise a regular simplex. Because all vertices are contained within the hyperplane with the equation
$$sum_{1le ile n+1}a_i x_i=1$$
it clearly is just $n$D. Its dihedral angle measured across the margins equally is given by $$arccos(1/n)$$
All edges of that regular simplex obviously have size $sqrt 2$.
When adjoining to the origin, then you'll still have a simplex, then sure being $(n+1)$D. But the slope of the lacing facets here is not so steep as would for the regular one (of according dimension). In fact the lacing margins would clearly show up dihedral angles of $90°$ each. - The size of the base edges here still is $sqrt 2$, while the size of the lacing edges, i.e. those connecting to the origin, clearly is just $1$.
--- rk
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
$endgroup$
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
add a comment |
$begingroup$
The vectors ${x_i}_{1le ile n+1}$ comprise a regular simplex. Because all vertices are contained within the hyperplane with the equation
$$sum_{1le ile n+1}a_i x_i=1$$
it clearly is just $n$D. Its dihedral angle measured across the margins equally is given by $$arccos(1/n)$$
All edges of that regular simplex obviously have size $sqrt 2$.
When adjoining to the origin, then you'll still have a simplex, then sure being $(n+1)$D. But the slope of the lacing facets here is not so steep as would for the regular one (of according dimension). In fact the lacing margins would clearly show up dihedral angles of $90°$ each. - The size of the base edges here still is $sqrt 2$, while the size of the lacing edges, i.e. those connecting to the origin, clearly is just $1$.
--- rk
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
$endgroup$
The vectors ${x_i}_{1le ile n+1}$ comprise a regular simplex. Because all vertices are contained within the hyperplane with the equation
$$sum_{1le ile n+1}a_i x_i=1$$
it clearly is just $n$D. Its dihedral angle measured across the margins equally is given by $$arccos(1/n)$$
All edges of that regular simplex obviously have size $sqrt 2$.
When adjoining to the origin, then you'll still have a simplex, then sure being $(n+1)$D. But the slope of the lacing facets here is not so steep as would for the regular one (of according dimension). In fact the lacing margins would clearly show up dihedral angles of $90°$ each. - The size of the base edges here still is $sqrt 2$, while the size of the lacing edges, i.e. those connecting to the origin, clearly is just $1$.
--- rk
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
edited Jan 17 at 15:29
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
answered Jan 17 at 8:07
Dr. Richard KlitzingDr. Richard Klitzing
1,71516
1,71516
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
add a comment |
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
$begingroup$
Please don't sign or initial posts. It's pointless (your full profile is already shown and linked under it) and makes them less clean.
$endgroup$
– Nij
Jan 18 at 22:00
add a comment |
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$begingroup$
The probability simplex is a face of the unit simplex. The unit simplex is the convex hull of the probability simplex and the origin.
$endgroup$
– Michael Burr
Jan 17 at 8:12