## Distribution of transformed multinomial variable?

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Suppose we have a uniform multinomial distribution over $2^d$ outcomes. Multinomial coefficients give distribution of vector valued variable $v$ where $v$ is the vector of observed counts.

Is there a combinatorial expression for the distribution of $D v$ where $D$ is a matrix obtained by $d$-fold Kronecker product of $A$, a $(-1,0,1)$ valued 2-by-2 matrix?

Three particular choices of $A$ I'm looking at are

$$A_1=\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)$$ $$A_2=\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$$ $$A_3=\left(\begin{matrix}1&0\\-1&1\end{matrix}\right)$$

Motivation: distribution of $Dv$ corresponds to distribution of feature vectors for three commonly feature bases for exponential families (Walsh, Amari's $\eta$ and Amari's $\theta$ bases respectively)

```d = 3;
walsh = KroneckerProduct @@ Table[{{1, 1}, {1, -1}}, {d}];
amariEta = KroneckerProduct @@ Table[{{1, 1}, {0, 1}}, {d}];
amariTheta = KroneckerProduct @@ Table[{{1, 0}, {-1, 1}}, {d}];
```

Edit: all 3 matrices are invertible, so we just need to multiply by appropriate inverse to get back vector of counts and corresponding multinomial coefficient. Inverse of $D$ is just the Kronecker product of inverses of corresponding base matrices

$$A_1^{-1}=\left(\begin{matrix}1/2&1/2\\1/2&-1/2\end{matrix}\right)$$ $$A_2^{-1}=\left(\begin{matrix}1&-1\\0&1\end{matrix}\right)$$ $$A_3^{-1}=\left(\begin{matrix}1&0\\1&1\end{matrix}\right)$$

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2
I'm voting to close as no longer relevant – Yaroslav Bulatov Sep 13 2010 at 17:55
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