Multi-objective MOCU (ηₘₒ)

• Multi-objective MOCU is a Bayesian measure that quantifies expected performance loss (regret) across multiple, potentially competing, objectives under model uncertainty.
• It generalizes single-objective MOCU by averaging regrets over uncertainty in both model parameters and decision-maker preferences through linear scalarization.
• It guides objective-based experimental design by ranking interventions to minimize uncertainty impact, as demonstrated in biological network applications.

The multi-objective mean objective cost of uncertainty (multi-objective MOCU, $\eta_{MO}$) is a Bayesian, objective-based quantification of uncertainty in systems governed by multiple, potentially competing, operational objectives. It measures the expected performance loss (regret) due to model uncertainty when selecting robust actions that optimize a user's (possibly unknown) preferences across several objectives. $\eta_{MO}$ generalizes the single-objective MOCU framework to multi-criteria scenarios critical for experiment design and operational decision-making in uncertain environments, such as biological networks, engineering systems, and multi-criteria optimization problems (Yoon et al., 2020).

1. Mathematical Definition and Derivation

Consider a parameter uncertainty class $\Theta$ with $\theta \in \Theta$ and a Bayesian posterior $p(\theta|D)$. Let $\mathcal{A}$ denote a finite or compact action/design space, and $J_1(\theta,a), \dots, J_m(\theta,a)$ represent $m$ operational objectives to be minimized. Let $w = (w_1, \dots, w_m)$ be a weight vector drawn from the $m$-simplex $W = \{ w : w_i \ge 0, \sum_i w_i = 1\}$ with distribution $p(w)$ (typically uniform).

Define the linear scalarized cost for model $\theta$ and action $a$:

$$J(\theta,a;w) \equiv \sum_{i=1}^m w_i J_i(\theta, a)$$

• Model-specific (clairvoyant) optimal action (if $\theta$ were known):

$$a_0(\theta, w) = \arg\min_{a \in \mathcal{A}} J(\theta, a; w)$$

• Robust (Bayes-optimal) action under uncertainty:

$$a^*(w) = \arg\min_{a \in \mathcal{A}} \mathbb{E}_{\theta \sim p(\theta|D)} [J(\theta, a; w)]$$

• Regret for model $\theta$ and weight $w$:

$$R(\theta, w) = J(\theta, a^*(w); w) - J(\theta, a_0(\theta, w); w) \geq 0$$

• Multi-objective MOCU averages the above regret:

$$\eta_{MO} = \mathbb{E}_{w \sim p(w)} \left[ \mathbb{E}_{\theta \sim p(\theta|D)} [R(\theta, w)] \right] = \int_W \int_\Theta (J(\theta, a^*(w);w) - J(\theta, a_0(\theta,w);w))\, p(\theta|D)\, d\theta\, p(w)\, dw$$

This scalar quantifies the expected cost incurred by not knowing the true model, averaged over both model parameter uncertainty and possible objective-weight (preference) uncertainty.

Derivation Sequence:

• Single-objective MOCU uses $a_0(\theta) = \arg\min_a \xi_\theta(a)$, $a^* = \arg\min_a \mathbb{E}_\theta[\xi_\theta(a)]$, and $$\eta = \mathbb{E}_\theta[\xi_\theta(a^*) - \xi_\theta(a_0(\theta))]$$.
• For $m=2$, scalarize via a parameter $\lambda \in [0,1]$ and average over $\lambda$.
• For general $m$, use $w \in W$, and the averaging proceeds over $p(w)$.

2. Underlying Assumptions

• Bayesian modeling: Parameter uncertainty encoded via a prior $\pi(\theta)$ and updated with new data $D$ to a posterior $p(\theta|D)$.
• Action/Objective space well-posedness: $\mathcal{A}$ and each $J_i(\theta,a)$ must be computable; global minima can be found.
• Linear scalarization: All objective trade-offs are representable through linear weighting.
• Weight distribution $p(w)$: Encodes preference among objectives, often set to uniform for objective indifference, but any Dirichlet prior is valid.
• No requirement for convexity or differentiability in objectives or action space, though these properties facilitate computation.

3. Computational Estimation of $\eta_{MO}$

The nested expectations in $\eta_{MO}$ do not generally admit closed-form solutions and are estimated via nested Monte Carlo:

1. Sample $w^{(\ell)}$, for $\ell = 1, \dots, L$, from $p(w)$.
2. For each $w^{(\ell)}$, sample $\theta^{(k)}$, $k = 1,\dots, K$, from $p(\theta|D)$.
3. For each $(w^{(\ell)}, \theta^{(k)})$, solve $$a_0^{(k,\ell)} = \arg\min_{a \in \mathcal{A}} J(\theta^{(k)}, a; w^{(\ell)})$$.
4. For each $w^{(\ell)}$, approximate the robust action: $$a^{*,(\ell)} \approx \arg\min_{a \in \mathcal{A}} \frac{1}{K} \sum_{k=1}^K J(\theta^{(k)}, a; w^{(\ell)})$$.
5. For each $\ell$, compute: $$\eta(w^{(\ell)}) \approx \frac{1}{K} \sum_{k=1}^K [J(\theta^{(k)}, a^{*,(\ell)}; w^{(\ell)}) - J(\theta^{(k)}, a_0^{(k,\ell)}; w^{(\ell)})]$$.
6. Estimate full $\eta_{MO}$ as $\frac{1}{L} \sum_{\ell=1}^L \eta(w^{(\ell)})$.

If the action space $\mathcal{A}$ is continuous or large, surrogate models, gradient-based optimization, genetic algorithms, or Bayesian optimization may be employed to solve the inner minimization problems.

4. Objective-Based Experimental Design Using $\eta_{MO}$

In Objective-based Experimental Design (OED), $\eta_{MO}$ is used to greedily select the next experiment $e \in E$ that maximally reduces the expected remaining uncertainty. For each candidate experiment:

1. Predict possible outcomes $y \sim p(y|D,e)$.
2. For each $y$, update posterior $p(\theta|D,e,y)$ and recompute $\eta_{MO}(D \cup \{e, y\})$ via Monte Carlo.
3. Estimate expected post-experiment MOCU: $$\mathrm{EMOCU}(e) \approx \mathbb{E}_y[\eta_{MO}(D \cup \{e,y\})]$$.
4. Select $e^* = \arg\min_e \mathrm{EMOCU}(e)$.

After observing $y^*$, update $D \leftarrow D \cup \{e^*, y^*\}$ and repeat. This process targets the experiment with greatest expected reduction in multi-objective uncertainty.

5. Theoretical Insights and Key Properties

• Nonnegativity: $\eta_{MO} \geq 0$ with equality if and only if, for all $w$ and $\theta$, the Bayes-optimal and clairvoyant actions coincide.
• Monotonicity under data accrual: Acquisition of new data, on average, decreases $\eta_{MO}$.
• Monte Carlo convergence: Estimation error decreases at rate $O(1/\sqrt{LK})$ for $L$ weight samples and $K$ model samples.
• Relation to Knowledge Gradient (KG): For Gaussian reward models and certain experimental classes, MOCU-based OED reduces to a KG policy.
• Approximate submodularity: In many scenarios, sequential experiment selection yields diminishing reductions in $\eta_{MO}$, though this is problem-dependent.

6. Mammalian Cell-Cycle Network Application

A Boolean Network with perturbation (BNp) comprising 10 mammalian cell-cycle genes (e.g., CycD, Rb, p27) serves as a real-world instance. Perturbation probability $p=0.01$; update via majority-rule Boolean logic. Some regulatory relationships are unknown, so $\theta$ indexes one of $2^u$ possible networks given $u$ unknown edges.

• Objectives:
  • $J_1$: Minimize steady-state probability $\pi_U$ that $(\mathrm{CycD} = \mathrm{Rb} = \mathrm{p27} = 0)$ (cancerous state).
  • $J_2$: Minimize probability $\pi_P$ that $(\mathrm{Cdc20}=1)$ but not in $U$, avoiding unintended pathology.
• Actions: Blockage of exactly one regulatory edge; thus, $|\mathcal{A}| =$ number of edges.

Procedure:

• Uniform prior over $\theta$ (network structures).
• Objective weight $\lambda \in [0,1]$, averaging two objectives: $$J(\theta,a;\lambda) = \lambda\pi_U + (1-\lambda)\pi_P$$; $\lambda \sim \mathrm{Uniform}[0,1]$.
• Monte Carlo sampling over $\theta$ and $\lambda$ to compute regrets.
• Produce line plots of $\eta_{MO}$ versus $u$ (number of unknown edges), summarizing over 500 random draws; $\eta_{MO}$ rises sharply with increased $u$, demonstrating heightened vulnerability to model uncertainty.

Impact: Ranking edges by their contribution to $\eta_{MO}$ enables targeted experimental efforts, guiding resolution of the most operationally significant uncertainties in regulatory structure.

7. Illustrative Examples and Figures

• Two-objective toy problem: For $$f(x;\lambda) = \lambda \alpha_1 (x-\gamma_1)^2 + (1-\lambda)\alpha_2(x-\gamma_2)^2$$, when $\gamma_1 = \gamma_2$ the optimum is $\lambda$-independent and MOCU is zero; otherwise, positive regret emerges, quantifying MOCU for the shift.
• Quadratic case studies: MOCU increases with both the interval $\Delta$ of uncertain parameters and shifts $(c,d)$ that affect the location of minima. Contrasted with entropy/variance, which fail to distinguish the operational impact of uncertainty.
• Mammalian cell-cycle network (Figure 1): $\eta_{MO}$ plotted against $u$, with shaded bands for min, mean + std, and median values over 500 randomizations, illustrating accelerating multi-objective cost with greater structural uncertainty.

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Multi-objective MOCU $\eta_{MO}$ encapsulates the operational impact of epistemic uncertainty in multi-objective settings, unifying Bayesian model uncertainty and decision preference uncertainty. Its computation involves nested sampling and optimization. Empirical results on diverse problem domains, including complex biological networks, demonstrate its value for objective-driven experiment prioritization and robust system intervention (Yoon et al., 2020).