On the arithmetic quasi depth

Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the quasi depth of $h$ by $qdepth(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$. We show that $qdepth(h)$ is a natural generalization for the quasi depth of a subposet $P\subset 2^{[n]}$ and we prove some basic properties of it. Given $h(j)=aj^p+b$, $j\geq 0$, with $a,b,n$ positive integers, we compute $qdepth(h)$ for $n=1,2$ and we give sharp bounds for $qdepth(h)$ for $p\geq 3$. Also, for $h(j)=a_nj^n+\cdots+a_1j+a_0$, $j\geq 0$, with $a_i\geq 0$, we prove that $qdepth(h)\leq 2^{n+1}$.Comment: 18 page

Toward Generalized Entropy Composition with Different q Indices and H-Theorem

An attempt is made to construct composable composite entropy with different $q$ indices of subsystems and address the H-theorem problem of the composite system. Though the H-theorem does not hold in general situations, it is shown that some composite entropies do not decrease in time in near-equilibrium states and factorized states with negligibly weak interaction between the subsystems.Comment: 25 pages, corrected some typos, to be published in J. Phys. Soc. Ja

Correlation bounds for fields and matroids

Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated for any distinct edges $e_1$ and $e_2$. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events $e \in B$, where $B$ is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of $k$-element independent sets of a matroid forms an ultra-log-concave sequence in $k$.Comment: 16 pages. Supersedes arXiv:1804.0307

P-values for high-dimensional regression

Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a `p-value lottery' and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error (FWER) and false discovery rate (FDR). In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially.Comment: 25 pages, 4 figure

Nonextensive Entropies derived from Form Invariance of Pseudoadditivity

The form invariance of pseudoadditivity is shown to determine the structure of nonextensive entropies. Nonextensive entropy is defined as the appropriate expectation value of nonextensive information content, similar to the definition of Shannon entropy. Information content in a nonextensive system is obtained uniquely from generalized axioms by replacing the usual additivity with pseudoadditivity. The satisfaction of the form invariance of the pseudoadditivity of nonextensive entropy and its information content is found to require the normalization of nonextensive entropies. The proposed principle requires the same normalization as that derived in [A.K. Rajagopal and S. Abe, Phys. Rev. Lett. {\bf 83}, 1711 (1999)], but is simpler and establishes a basis for the systematic definition of various entropies in nonextensive systems.Comment: 16 pages, accepted for publication in Physical Review