Bayesian Shannon-code evidence rate

Let $S$ consist of a positive finite number of parties, positive finite numbers of settings for each party, and positive finite numbers of outcomes at each party and setting. Write $A$ for its joint settings, $X_a$ for its joint outcomes at setting $a$, $V$ for the finite set of assignments prescribing an outcome at every setting of every party, and $v|_a$ for the prescribed joint outcome at $a$. Let $Q(a,x)$ be a real table such that each $Q(a,\cdot)$ is nonnegative and sums to $1$; for every subset $J$ of parties, settings $a,b$ agreeing on $J$, and $v\in V$, the sum of $Q(a,x)$ over $x_i=v_i(a_i)$ for all $i\in J$ equals the corresponding sum of $Q(b,x)$ over $x_i=v_i(b_i)$; and there exist positive local dimensions $d_i$, a positive semidefinite joint matrix $\rho$ of trace $1$, and positive semidefinite local matrices $M_{i,a_i,x_i}$, summing to the identity over $x_i$, such that $Q(a,x)=\operatorname{Re}\operatorname{tr}(E_{a,x}\rho)$, with $E_{a,x}(u,v)=\prod_iM_{i,a_i,x_i}(u_i,v_i)$. Let $s$ be any nonnegative distribution of total mass $1$ on $A$, including distributions with zero coordinates. For a distribution $p\in\mathcal P(V)$, define $P_p(a,x)=\sum_{v:\,v|_a=x}p(v)$, and define trial laws on $B=\{(a,x):a\in A,\ x\in X_a\}$ by $q(a,x)=s(a)Q(a,x)$ and $r_p(a,x)=s(a)P_p(a,x)$. Define $D(q\|r)=+\infty$ if some trial has $q>0$ and $r=0$, and otherwise define it as the extended nonnegative real corresponding to $\max\{0,\sum_{b:q(b)\ne0}q(b)\log_2(q(b)/r(b))\}$; put $\Delta=\inf_{p\in\mathcal P(V)}D(q\|r_p)$. Let $\nu$ be Hausdorff measure of dimension $|V|-1$, restricted to $\mathcal P(V)$, and let $w:\mathbb R^V\to\mathbb R$ be continuous on that simplex, satisfy $c<w(p)$ throughout it for some real $c>0$, and have $\int w\,d\nu=1$. For each length-$n$ trial sample $t$, put $L_q(t)=\prod_iq(t_i)$ and $m_n(t)=\int w(p)\prod_i r_p(t_i)\,d\nu(p)$. Using the finite enumeration $e_n$ supplied on the length-$n$ sample space, define

and let $W_n(t)$ be the Boolean word of length $\ell_n(t)$ whose entry at index $j$ is true exactly when $\lfloor2^{j+1}c_n(t)\rfloor_+\bmod2=1$; natural-number floor and ceiling are truncated at zero, and the total logarithm has $\log_2 0=0$. Let $\mathbb P_q$ be the infinite product of the trial measure $\sum_{b\in B}\max\{q(b),0\}\delta_b$. The conclusion is that $w$ is integrable with respect to $\nu$, $\mathbb P_q$ has total mass $1$, $\Delta<+\infty$, and, for every $n$, $W_n$ restricted to samples with $m_n(t)>0$ is prefix-free, meaning that a prefix relation between two such words implies equality of their samples. Moreover, for every $n$ and every sample $t$ with $m_n(t)>0$,

Finally, for $\mathbb P_q$-almost every infinite sequence $z$,

where $z_{<n}=(z_0,\ldots,z_{n-1})$ and $\Delta_{\mathbb R}$ is the real value of the finite number $\Delta$. All universal claims about $n$ include $n=0$: its likelihood and marginal are $1$, its word is empty, and the quotient's term at $n=0$ is $0$ under total real division.