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→Active inference and optimal control 主动推理与最优控制
Active inference is related to [[optimal control]] by replacing value or cost-to-go functions with prior beliefs about state transitions or flow.<ref>Friston, K., (2011). [http://www.fil.ion.ucl.ac.uk/~karl/What%20Is%20Optimal%20about%20Motor%20Control.pdf What is optimal about motor control?]. Neuron, 72(3), 488–98.</ref> This exploits the close connection between Bayesian filtering and the solution to the [[Bellman equation]]. However, active inference starts with (priors over) flow <math> f = \Gamma \cdot \nabla V + \nabla \times W </math> that are specified with scalar <math> V(x) </math> and vector <math> W(x) </math> value functions of state space (c.f., the [[Helmholtz decomposition]]). Here, <math> \Gamma </math> is the amplitude of random fluctuations and cost is <math> c(x) = f \cdot \nabla V + \nabla \cdot \Gamma \cdot V</math>. The priors over flow <math> p(\tilde{x}\mid m) </math> induce a prior over states <math> p(x\mid m) = \exp (V(x)) </math> that is the solution to the appropriate forward [[Kolmogorov equations]].<ref>Friston, K., & Ao, P. (2012). [http://www.fil.ion.ucl.ac.uk/~karl/Free%20Energy%20Value%20and%20Attractors.pdf Free-energy, value and attractors]. Computational and mathematical methods in medicine, 2012, 937860.</ref> In contrast, optimal control optimises the flow, given a cost function, under the assumption that <math> W = 0 </math> (i.e., the flow is curl free or has detailed balance). Usually, this entails solving backward [[Kolmogorov equations]].<ref>Kappen, H., (2005). [https://arxiv.org/abs/physics/0505066 Path integrals and symmetry breaking for optimal control theory]. Journal of Statistical Mechanics: Theory and Experiment, 11, p. P11011.</ref>
Active inference is related to [[optimal control]] by replacing value or cost-to-go functions with prior beliefs about state transitions or flow.<ref>Friston, K., (2011). [http://www.fil.ion.ucl.ac.uk/~karl/What%20Is%20Optimal%20about%20Motor%20Control.pdf What is optimal about motor control?]. Neuron, 72(3), 488–98.</ref> This exploits the close connection between Bayesian filtering and the solution to the [[Bellman equation]]. However, active inference starts with (priors over) flow <math> f = \Gamma \cdot \nabla V + \nabla \times W </math> that are specified with scalar <math> V(x) </math> and vector <math> W(x) </math> value functions of state space (c.f., the [[Helmholtz decomposition]]). Here, <math> \Gamma </math> is the amplitude of random fluctuations and cost is <math> c(x) = f \cdot \nabla V + \nabla \cdot \Gamma \cdot V</math>. The priors over flow <math> p(\tilde{x}\mid m) </math> induce a prior over states <math> p(x\mid m) = \exp (V(x)) </math> that is the solution to the appropriate forward [[Kolmogorov equations]].<ref>Friston, K., & Ao, P. (2012). [http://www.fil.ion.ucl.ac.uk/~karl/Free%20Energy%20Value%20and%20Attractors.pdf Free-energy, value and attractors]. Computational and mathematical methods in medicine, 2012, 937860.</ref> In contrast, optimal control optimises the flow, given a cost function, under the assumption that <math> W = 0 </math> (i.e., the flow is curl free or has detailed balance). Usually, this entails solving backward [[Kolmogorov equations]].<ref>Kappen, H., (2005). [https://arxiv.org/abs/physics/0505066 Path integrals and symmetry breaking for optimal control theory]. Journal of Statistical Mechanics: Theory and Experiment, 11, p. P11011.</ref>
主动推理与[[最优控制]]有关，它用状态转移或流的先验信念替换价值或成本函数。<ref>Friston, K., (2011). [http://www.fil.ion.ucl.ac.uk/~karl/What%20Is%20Optimal%20about%20Motor%20Control.pdf What is optimal about motor control?]. Neuron, 72(3), 488–98.</ref>这充分利用了贝叶斯滤波和[[Bellman方程]]解之间的紧密联系。然而，主动推理从状态空间的标量<math>V（x）</math>和向量<math>W（x）</math>值函数（c.f.，Helmholtz分解）指定的流<math>f=\Gamma\cdot\nabla V+\nabla\times W</math>开始。这里，<math>\Gamma</math>是随机波动的幅度，成本是<math>c（x）=f\cdot\nabla V+\nabla\cdot\Gamma\cdot V</math>。流上的先验<math>p（\tilde{x}\mid m）</math>诱导了一个先验的超状态<math>p（x\mid m）=\exp（V（x））</math>这是相应的正向[[Kolmogorov方程]]的解。<ref>Friston, K., & Ao, P. (2012). [http://www.fil.ion.ucl.ac.uk/~karl/Free%20Energy%20Value%20and%20Attractors.pdf Free-energy, value and attractors]. Computational and mathematical methods in medicine, 2012, 937860.</ref>相反，在假设<math>W=0的情况下，最优控制优化了给定成本函数的流量（即，流量没有旋度或具有详细平衡）。通常，这需要向后求解[[Kolmogorov方程]]。<ref>Kappen, H., (2005). [https://arxiv.org/abs/physics/0505066 Path integrals and symmetry breaking for optimal control theory]. Journal of Statistical Mechanics: Theory and Experiment, 11, p. P11011.</ref>
=== Active inference and optimal decision (game) theory 主动推理与最优决策（博弈）理论===
=== Active inference and optimal decision (game) theory 主动推理与最优决策（博弈）理论===