13 problems

Let $\rho$ range over qubit states parametrized by a Bloch-vector radius $r\in[0,1]$, let $\zeta_n(u)$ be the corresponding $n$-copy quantum coding state in the restricted family…

Let the quantum Jeffreys' prior for the qubit density matrices be the normalized prior … The classical Jeffreys' prior is known to yield the asymptotic common minimax and maximin r…

Let a prequential model be constructed from an arbitrary in-model estimator, and let $P$ be the data-generating distribution. Variance-independent redundancy conjecture. Regardless…

Consider a one-parameter model with data-generating distribution $P$ and model parameter $\mu^*$, and let an in-model estimator be used to construct a prequential model. In-model r…

Let $\mathcal X$ be a sample space, and let $P$, $M_{\mu}$, and $\mu^*$ be as in the paper's main theorem, with $\mu^*$ the model parameter associated with the distribution in the…

Landscape criterion. We conjecture that, for any additively optimal UTM, a good UFCC $U$ should have a landscape—that is, an ordering of codebooks—similar to the UTM's landscape in…

Prefix-free penalty conjecture. Universal prefix-free codes incur, compared with the optimal rate in equation, a penalty of at best

Optimal-rate conjecture. The rate obtained by this universal one-to-one compression scheme is optimal; equivalently, a matching converse should hold for the rate above.

Let $P\in\mathcal{P}_{k}$, and let $\mathcal{F}^{\rm{\scriptscriptstyle{(TU)}}}_n$ be the twice-universal fixed-to-variable random number generator described above, with output alp…

Hilberg's conjecture. The mutual information between two adjacent blocks of length $n$ satisfies

Let $d\geq 2$ and let $M(\Delta,d)$ denote the penalty function defined by the minimization over pairs of distributions with entropy difference $\Delta$. For $\Delta\in[0,\log d]$,…

Consider the universal communication setting of Theorem … can be extended to the source. The authors point to ideas of Ziv and Ziv2 as potentially relevant; the source gives no res…

Let $\mathcal{M}$ be a $k$-dimensional parametric model, and let a prequential plug-in code use estimators constrained to lie in $\mathcal{M}$. Let data be i.i.d. according to $P$,…