贝叶斯推断常用于参数估计，但后验分布可能不足以支持决策。贝叶斯摊销决策方法通过学习数据和动作对的成本，做出贝叶斯最优决策。该方法在模拟推理基础上，针对决策任务引入成本函数，旨在最小化不确定条件下的期望成本。文章提出了一种贝叶斯摊销决策方法（BAM），利用神经网络回归数据和动作对的成本，从而避免计算后验分布并重复评估成本函数，最终实现高效的贝叶斯决策。通过与蒙特卡洛方法的对比实验，BAM在模拟资源消耗方面展现出显著优势，并成功应用于医学神经科学等实际场景。

🤔 **贝叶斯决策目标：** 在不确定条件下，选择能够最小化期望成本的动作，其中成本函数量化了在已知系统真实参数的情况下采取特定动作的代价。

💡 **贝叶斯摊销决策（BAM）方法：** BAM通过学习数据和动作对的成本，直接估计期望成本，避免了计算后验分布并重复评估成本函数。它将贝叶斯决策建模为一个回归任务，利用神经网络回归参数集的成本。

📊 **成本函数定义：** 成本函数通常将真实参数和动作的偏差作为惩罚项，例如，在流行病学中，成本函数可以包含疫苗接种、隔离或封锁的经济成本。

📈 **实验结果：** 与基于蒙特卡洛的贝叶斯决策方法相比，BAM在达到相同决策质量的情况下，显著减少了模拟资源的消耗，体现了其在效率方面的优势。

🧪 **应用场景：** BAM 已成功应用于Lotka-Volterra模型、SIR模型等基准任务，以及医学神经科学等实际场景，展现了其在不同领域的应用潜力。

Bayesian inference is a popular tool for parameter estimation. However, the posterior distribution might not be sufficient for decision-making. Bayesian Amortized Decision-Making is a method that learns the cost of data and action pairs to make Bayes-optimal decisions.Simulation-BasedInference (SBI) is apowerful tool for estimating the posterior distribution $p(\theta \mid x)$ overthe parameters $\theta$ of a simulator, given observed data $x$. However, thegoal is sometimes not the posterior itself but but making a decision in adownstream task based on the inferred posterior distribution. Often, thesedecisions are associated with a cost, which one wishes to minimize. This is whereBayesian decision-making comes into play, aiming to choose actions that minimizethe expected cost under uncertain conditions.Approximate Bayesian Decision MakingGiven an observation $x_o$ and a posterior $p(\theta \mid x)$, Bayesian decisionmaking provides the action with the lowest cost, averaged over thedistribution of parameters.$$a = \underset{a \in \mathcal{A}}{\operatorname{arg\ min}} \int c(\theta, a) p(\theta \mid xo) d\theta$$The function $c(\theta, a)$ quantifies the cost of taking an action $a$ if thetrue parameters $\theta$ of the system were known but is flexible enough toallow for different cost structures as well. The true posterior $p(\theta \midx)$ is usually not known and is approximated using SBI, usually by a conditionaldensity estimator $q{\phi}(\theta \mid x)$. The quality of the decision hingesthen on the accuracy of this posterior approximation.Figure 1. [Gor23A], Figure 1. Illustration of the difference betweenthe proposed method for amortized Bayesian decision making and numericalapproximation of the integral over the cost function, weighted by the estimatedposterior. Each marker describes the averaged cost difference for tenobservations.To address this challenge, [Gor23A] introduce Bayesian Amortized Decision Making (BAM) inthe context of SBI. Within the same setting as neural posterior estimation(NPE), BAM learns the cost of data and action pairs. Instead of averaging thecost over the posterior, the proposed method requires only one forward-passthrough the network. Therefore, BAM performs amortized Bayesian decision-making.BAM aims to estimate the expected cost $\mathbb{E}_{p(\theta | xo)}[c(\theta,a)]$ under the true posterior. This is achieved by samplingfrom the joint distribution $(\theta, x) \sim p(\theta, x)$ and an actiondistribution $a \sim p(a)$. A feedforward neural network $f{\omega}(x,a)$ is then used to regressthe cost of parameter sets, utilizing a Mean-Squared-Error loss.$$\mathcal{L}(\omega) = \mathbb{E}{p(\theta, a)p(a)} \left[ \left(f{\omega}(x,a) - c(\theta, a)\right)^2 \right]$$To define the cost function, the authors assign zero cost where $a =\theta{\text{true}}$ and increase the cost the more $a$ deviates from$\theta{\text{true}}$. The exact manner in which the deviation is penalizeddepends on the task at hand. In real-world scenarios, the cost function couldalso include the economic cost of the action. Taking epidemiology as an example,the cost function could include a cost for vaccination, quarantine, or even alockdown.The authors prove that BAM accurately yields the expected cost and framesBayesian decision making as a regression task, offering an efficient alternativeto weighted parameter selection. In contrast to a Monte-Carlo approximation ofthe integral (NPE-MC) with samples from the (approximate) posterior$$\mathbb{E}_{p(\theta | xo)}[c(\theta, a)] \approx \frac{1}{N} \sum{i=1}^{N} c(\thetai, a).$$BAM directly learns the expected cost $f{\omega}(x,a)$. It thereby circumventsthe need to learn the full posterior distribution and repeatedly evaluating thecost function $c(\theta, a)$.Numerical ExperimentsFigure 2. [Gor23A], Figure 3. Comparison of the proposed BAM method andthe Monte-Carlo based approximation of the expected cost against the incurredcost using the true posterior of each task.To illustrate the effectiveness and limitations of BAM, the authors compare itwith a Monte-Carlo-based approach (NPE-MC) across various benchmark taskstypical to SBI, such as the Lotka-Volterra and SIR models. Additionally, theypresent an application to a real-world scenario in medical neuroscience. Thecomparison on the SBI tasks (Figure 2), based on six differentsimulation budgets, reveals that the Monte-Carlo variant necessitates a largersample size to achieve a quality of solution comparable to BAM, suggestingsignificant savings in simulation resources when only Bayes-optimal solutionsare sought.