--- title: Bidirectional Age of Incorrect Information (BAoII) url: https://www.emergentmind.com/topics/bidirectional-age-of-incorrect-information-baoii type: topic --- # Bidirectional Age of Incorrect Information (BAoII) Bidirectional Age of Incorrect Information (BAoII) is a performance metric for status updates in virtual dynamic environments (VDEs) such as the Metaverse and digital twins (DTs), where interacting entities must maintain mutual awareness of themselves and of one another. It extends the Age of Incorrect Information (AoII) from a unidirectional setting to bidirectional information exchange by quantifying the time-dependent penalty incurred when an entity has incorrect or outdated knowledge about itself, the other entity, or the other entity’s knowledge of it. In the formulation introduced for VDEs, BAoII is analyzed through a continuous-time Markov chain (CTMC), yielding a closed-form long-term expression and a transmission-cost threshold that characterizes optimal update strategies [2507.13312]. ## 1. Conceptual foundations BAoII is motivated by interactive settings in which correctness is not exhausted by the statement “the latest packet is recent.” In a VDE, each entity has a true state, a self-view derived from local sensing, and a remote view of the other entity derived from received packets. From the viewpoint of entity 1, the information configuration relevant to BAoII is incorrect when at least one of the following is wrong: entity 1’s knowledge of its own state, entity 1’s knowledge of entity 2’s state, or entity 2’s knowledge of entity 1’s state. The metric therefore couples self-awareness, awareness of the other, and reciprocated awareness in a single freshness-correctness construct [2507.13312]. This coupling distinguishes BAoII from classical Age of Information (AoI) and from standard AoII. AoI is a unidirectional age metric, typically written as $\Delta(t)=t-u(t)$, and does not distinguish whether the information is still correct. AoII penalizes only periods during which the estimate is wrong, with the penalty growing since the information became incorrect. BAoII generalizes this idea to a bidirectional interaction: reset is governed by collective correctness conditions rather than by a single source-monitor relation [2507.13312]. A common misunderstanding is to treat BAoII as merely “two AoII processes, one in each direction.” The original formulation rejects that interpretation. For a given entity, reset depends on a joint correctness condition across multiple knowledge components, so the metric is stricter than a simple sum of two independent unidirectional incorrectness ages. This suggests a metric tailored to co-manipulation, immersive collaboration, and DT-based control, where “I know where you are” and “you know where I am” are operationally coupled [2507.13312]. ## 2. System model and CTMC representation The basic model contains two entities, indexed by $i\in\{1,2\}$. Their true states are $X_1(t)$ and $X_2(t)$. Each entity has a self-estimate, $\widehat{X}_1(t)$ and $\widehat{X}_2(t)$, obtained from local measurements, and a remote estimate, $\widetilde{X}_1(t)$ and $\widetilde{X}_2(t)$, obtained from packets sent by the other side. Time is slotted with normalized slot duration $1$, and the analytical development subsequently uses a CTMC representation [2507.13312]. Correctness is abstracted into binary indicators: \[ \widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t), \] where $\oplus$ is bitwise XOR. A value of $0$ means correct knowledge and a value of $1$ means incorrect knowledge. The global information state is the four-bit vector \[ [\widehat{Y}_1(t),\widetilde{Y}_1(t),\widehat{Y}_2(t),\widetilde{Y}_2(t)]\in\{0,1\}^4. \] Each entity measures its own state with rate $m$ and transmits with rate $\lambda=pm$, where $p\in[0,1]$. Transmissions are always tied to a measurement event: “measure only” occurs at rate $(1-p)m$, while “measure + transmit” occurs at rate $pm$. Drift in the true state occurs with rate $d$ and makes the corresponding self-view incorrect; because the other entity’s remote view then also becomes stale, drift imposes logical constraints on reachable four-bit configurations [2507.13312]. Two classes of four-bit states are impossible: configurations of the form $[1,*,*,0]$ and $[*,0,1,*]$. After excluding impossible combinations and grouping the remaining configurations, the CTMC has nine states. The state space is \[ \mathcal{S}=\{\mathrm{O},\Phi,\mathrm{A},\mathrm{B},\Gamma,\mathrm{F},\Psi,\Theta,\mathrm{E}\}. \] | State | Coding | Brief meaning | |---|---|---| | $\mathrm{O}$ | $[0,0,0,0]$ | complete correct information for both | | $\Phi$ | $[0,0,0,1]$ | entity 1 fully correct; entity 2 wrong about entity 1 | | $\mathrm{A}$ | $[1,0,0,1]$ | entity 1 wrong on self; entity 2 wrong about entity 1 | | $\mathrm{B}$ | $[0,1,1,0]$ | each entity correct on self, wrong about the other | | $\Gamma$ | $[0,1,1,1]$ | entity 1 wrong about entity 2; entity 2 in complete error | | $\mathrm{F}$ | $[0,1,0,1]$ | both correct on self, each wrong about the other | | $\Psi$ | $[1,1,0,1]$ | entity 1 wrong on self and on entity 2 | | $\Theta$ | $[0,1,0,0]$ | entity 1 wrong about entity 2; entity 2 fully correct | | $\mathrm{E}$ | $[1,1,1,1]$ | complete error for both | The CTMC transitions are induced only by drift and by measurement with or without immediate transmission. In the symmetric case, the measurement processes are independent Poisson processes with common rate $m$, the drifts are independent Poisson processes with rate $d$, and the transmission probability $p$ is common to both entities [2507.13312]. ## 3. Formal definition of BAoII For entity 1, BAoII is defined as a time-dependent penalty \[ \Delta_{\mathrm{err}}^{(1)}(t)=\big(\widehat{Y}_1(t)\lor \widehat{Y}_2(t)\lor \widetilde{Y}_1(t)\big)\,f^{(1)}(t), \] where $\lor$ is logical OR and $f^{(1)}(t)$ is an increasing penalty function. The logical structure is the core novelty: BAoII is zero only when all three relevant correctness bits are zero, and it is positive whenever any one of them is incorrect [2507.13312]. The paper adopts a linear penalty function, \[ f^{(1)}(t)=t-\sigma^{(1)}(t), \] with \[ \sigma^{(1)}(t)=\max\{t'\le t:\widehat{Y}_1(t')=\widehat{Y}_2(t')=\widetilde{Y}_1(t')=0\}. \] Thus BAoII for entity 1 grows linearly with the elapsed time since the last instant at which entity 1 knew itself correctly, entity 2 knew itself correctly, and entity 1 knew entity 2 correctly. This makes the penalty a sawtooth process: it ramps during error periods and resets when the required collective correctness condition is restored [2507.13312]. The reset structure is viewpoint-dependent. For entity 1, the reset states are $\mathrm{O}$ and $\Phi$, because both satisfy $\widehat{Y}_1=0$, $\widehat{Y}_2=0$, and $\widetilde{Y}_1=0$. In $\Phi$, only entity 2’s knowledge of entity 1 is wrong, so $\Phi$ is a reset state for entity 1’s BAoII but not a full-system mutual-awareness state. Full system-wide correctness corresponds to $\mathrm{O}=[0,0,0,0]$ [2507.13312]. This distinction matters conceptually. BAoII is bidirectional, but it is not purely global. It can be instantiated from the viewpoint of a specific entity while still preserving the coupling between both directions of interaction. A plausible implication is that application-level optimization may need to distinguish between per-entity mutual-awareness objectives and stricter system-wide consistency objectives. ## 4. Closed-form behavior and analytical results The CTMC uses the rate matrix $\mathbf{Q}=\{q_{hn}\}_{h,n\in\mathcal{S}}$, with off-diagonal entries given by drift and measurement/transmission events and diagonal terms satisfying $q_{hh}=-\sum_{n\ne h}q_{hn}$. Under ergodicity, the stationary distribution $\boldsymbol{\pi}$ satisfies \[ \boldsymbol{\pi}\mathbf{Q}=\mathbf{0},\qquad \sum_{s\in\mathcal{S}}\pi_s=1. \] For the reset states, the paper reports \[ \pi_{\Phi}= \frac{d m_1 m_2 (1-p)^2 p} {(d+m_1(1-p)p)(d+m_2(1-p)p)(d+m_1-m_1p^2)}, \] \[ \pi_{\mathrm{O}}= \frac{m_1 m_2 (1-p)^2 p^2} {(d+m_1(1-p)p)(d+m_2(1-p)p)}. \] Conditioned on being in one of the reset states, the renewal-cycle starting probabilities are \[ P_{\mathrm{O}}=\frac{\pi_{\mathrm{O}}}{\pi_{\mathrm{O}}+\pi_{\Phi}},\qquad P_{\Phi}=\frac{\pi_{\Phi}}{\pi_{\mathrm{O}}+\pi_{\Phi}}. \] From $\mathrm{O}$ or $\Phi$, the first transition into an error state occurs only via drift, leading to the first-error states $\mathrm{A}$, $\mathrm{B}$, or $\Gamma$ [2507.13312]. The paper then introduces expected hitting times $\tau_s$ to return to a reset state from each non-reset state, together with conditional jump probabilities \[ s_{h,n}=\frac{q_{hn}}{\sum_{l\neq h}q_{hl}}. \] These yield a linear system of standard Markov-chain hitting-time equations, with $\tau_{\mathrm{O}}=\tau_{\Phi}=0$. One explicit relation reported in the appendix is \[ \tau_{\mathrm{A}}=\frac{2}{d+m_1}+\frac{d}{d+m_1}\tau_{\mathrm{E}}. \] After solving the system and averaging over the first-error mixture, the mean error-cycle duration is \[ \mathcal{T}=\frac{1}{m_1}+\frac{1}{2pm_2}, \] which becomes, in the symmetric case $m_1=m_2=m$, \[ \mathcal{T}=\frac{1}{m}+\frac{1}{2pm} =\frac{2p+1}{2pm}. \] The long-term BAoII is then \[ \Delta_{\mathrm{BAoII}}=\frac{\mathcal{T}}{2} =\frac{2p+1}{4pm}. \] This closed form decreases with increasing $m$ and with increasing $p$ [2507.13312]. One notable feature is the absence of the drift rate $d$ from the final symmetric expression. This does not mean drift is irrelevant; rather, the paper’s interpretation is that drift determines how often error periods are triggered, whereas the average duration of an error episode depends on how quickly measurements and transmissions resolve the episode. A common misconception is therefore to expect $d$ to appear explicitly in the closed-form BAoII expression; in this model, it does not [2507.13312]. ## 5. Cost-sensitive optimization and design implications To couple correctness-freshness performance with resource expenditure, the model assigns a measurement cost $k_m$ and a transmission cost $k_{\lambda}$. Since measurements occur at rate $m$ and transmissions at rate $\lambda=pm$, the average per-unit-time resource cost is \[ K=k_m m+k_{\lambda}pm. \] The total cost is defined as \[ \mathcal{C}=\Delta_{\mathrm{BAoII}}+K, \] which, in the symmetric case, becomes \[ \mathcal{C}(p,m)=\frac{2p+1}{4pm}+(k_m+p k_{\lambda})m. \] This makes the central design trade-off explicit: increasing $m$ and $p$ reduces BAoII but increases resource expenditure [2507.13312]. The key optimization result is a transmission-cost threshold. There exists \[ k_{\lambda}^{\circ}=\frac{1}{4m^2} \] such that, if $k_{\lambda}