# 自由能原理

{{简介{神经科学中由卡尔J.弗里斯顿提出的假说}}

## 定义

Definition (continuous formulation): Active inference rests on the tuple $\displaystyle{ (\Omega,\Psi,S,A,R,q,p) }$,

Definition (continuous formulation): Active inference rests on the tuple $\displaystyle{ (\Omega,\Psi,S,A,R,q,p) }$,

“Definition”（连续公式）：主动推理基于元组$\displaystyle{ （\Omega，\Psi，S，A，R，q，p） }$

• A sample space $\displaystyle{ \Omega }$ – from which random fluctuations $\displaystyle{ \omega \in \Omega }$ are drawn
• “一个样本空间”$\displaystyle{ \Omega }$–从中提取随机波动$\displaystyle{ \Omega\in\Omega }$
• Hidden or external states $\displaystyle{ \Psi:\Psi\times A \times \Omega \to \mathbb{R} }$ – that cause sensory states and depend on action
• “隐藏或外部状态”$\displaystyle{ \Psi:\Psi\times A\times\Omega\to\mathbb{R} }$——引起感觉状态并依赖于动作
• Sensory states $\displaystyle{ S:\Psi \times A \times \Omega \to \mathbb{R} }$ – a probabilistic mapping from action and hidden states
• “感觉状态”$\displaystyle{ S:\Psi\times A\times\Omega\to\mathbb{R} }$——动作和隐藏状态的概率映射
• Action $\displaystyle{ A:S\times R \to \mathbb{R} }$ – that depends on sensory and internal states
• “动作”$\displaystyle{ A:S\times R \to \mathbb{R} }$——这取决于感觉和内部状态
• Internal states $\displaystyle{ R:R\times S \to \mathbb{R} }$ – that cause action and depend on sensory states
• “内部状态”$\displaystyle{ R:R\times S\to\mathbb{R} }$——引起动作并依赖于感觉状态
• Generative density $\displaystyle{ p(s, \psi \mid m) }$ – over sensory and hidden states under a generative model $\displaystyle{ m }$
• “生成密度”$\displaystyle{ p(s, \psi \mid m) }$——在生成模型下的感觉和隐藏状态
• Variational density $\displaystyle{ q(\psi \mid \mu) }$ – over hidden states $\displaystyle{ \psi \in \Psi }$ that is parameterised by internal states $\displaystyle{ \mu \in R }$
• “变分密度”$\displaystyle{ q(\psi \mid \mu) }$–由R中的内部状态$\displaystyle{ \mu \in R }$参数化的隐藏状态$\displaystyle{ \psi \in \Psi }$

### Action and perception 行动与感知

$\displaystyle{ \lim_{T\to\infty} \frac{1}{T} \underset{\text{free-action}} {\underbrace{\int_0^T F(s(t),\mu (t))\,dt}} \ge \lt math \gt lim { t to infty } frac {1}{ t } underset { text { free-action }{ underbrace { int _ 0 ^ t f (s (t) ，mu (t)) ，dt } ge 目标是最大化模型证据\lt math\gt p(s\mid m) }$或最小化意外$\displaystyle{ -\log p(s\mid m) }$。这通常涉及隐藏态的难以处理的边缘化，因此意外被一个较高的变分自由能边界所取代。[9]然而，这意味着内部状态也必须最小化自由能，因为自由能是感官和内部状态的函数：


\lim_{T\to\infty} \frac{1}{T} \int_0^T \underset{\text{surprise}}{\underbrace{-\log p(s(t)\mid m)}} \, dt = H[p(s\mid m)] [/itex]

$\displaystyle{ a(t) = \underset{a}{\operatorname{arg\,min}} \{ F(s(t),\mu(t)) \} }$

$\displaystyle{ \mu(t) = \underset{\mu}{\operatorname{arg\,min}} \{ F(s(t),\mu)) \} }$

$\displaystyle{ \underset{\mathrm{free-energy}} {\underbrace{F(s,\mu)}} = \underset{\mathrm{energy}} {\underbrace{ E_q[-\log p(s,\psi \mid m)]}} - \underset{\mathrm{entropy}} {\underbrace{ H[q(\psi \mid \mu)]}} = \underset{\mathrm{surprise}} {\underbrace{ -\log p(s \mid m)}} + \underset{\mathrm{divergence}} {\underbrace{ D_{\mathrm{KL}}[q(\psi \mid \mu) \parallel p(\psi \mid s,m)]}} 所有的贝叶斯推断都可以用自由能最小化来表达，例如，当自由能相对于内态最小化时，隐态上变分密度和后验密度之间的Kullback-Leibler散度最小化。当变分密度的形式固定时，这对应于近似贝叶斯推理，反之则对应于精确贝叶斯推理。因此，自由能最小化提供了贝叶斯推理和滤波（如Kalman滤波）的一般描述。复杂度和贝叶斯模型可以有效地分解为自由能量选择： \geq \underset{\mathrm{surprise}} {\underbrace{ -\log p(s \mid m)}} }$

$\displaystyle{ \underset{\text{free-energy}} {\underbrace{ F(s,\mu)}} = \underset{\text{complexity}} {\underbrace{ D_\mathrm{KL}[q(\psi\mid\mu)\parallel p(\psi\mid m)]}} - \underset{\mathrm{accuracy}} {\underbrace{E_q[\log p(s\mid\psi,m)]}} }$


## Free energy minimisation 自由能最小化

### Free energy minimisation and self-organisation 自由能最小化和自组织

$\displaystyle{ \lim_{T\to\infty} \frac{1}{T} \underset{\text{free-action}} {\underbrace{\int_0^T F(s(t),\mu (t))\,dt}} \ge \lim_{T\to\infty} \frac{1}{T} \int_0^T \underset{\text{surprise}}{\underbrace{-\log p(s(t)\mid m)}} \, dt = H[p(s\mid m)] }$

### Free energy minimisation and Bayesian inference 自由能最小化与贝叶斯推理

$\displaystyle{ \underset{\text{free-energy}} {\underbrace{ F(s,\mu)}} = \underset{\text{complexity}} {\underbrace{ D_\mathrm{KL}[q(\psi\mid\mu)\parallel p(\psi\mid m)]}} - \underset{\mathrm{accuracy}} {\underbrace{E_q[\log p(s\mid\psi,m)]}} }$

$\displaystyle{ \dot{\tilde{\mu}} = D \tilde{\mu} - \partial_{\mu}F(s,\mu)\Big|_{\mu = \tilde{\mu}} }$


## Free energy minimisation in neuroscience 神经科学中的自由能最小化

### Perceptual inference and categorisation 感性推理与分类

$\displaystyle{ \dfrac{\partial E^{total}(Y^{VP},X^{SN},x^{CN},y^{KN})}{\partial y^{SN}_{mn}}=x^{CN}_{mn}-b^{CN}\varepsilon^{CN}_{nm}+b^{CN}\sum_{k}(\varepsilon^{KN}_{knm}) }$

(y ^ { VP } ，x ^ { SN } ，x ^ { CN } ，y ^ { KN }){ partial y ^ { SN }{ mn }}} = x ^ { CN }-b ^ { CN } varepsilon ^ { nm } + b ^ { CN } sum { k }(varepsilon ^ { KN }{ m }) </knmath >

$\displaystyle{ \dot{\tilde{\mu}} = D \tilde{\mu} - \partial_{\mu}F(s,\mu)\Big|_{\mu = \tilde{\mu}} }$

### Perceptual precision, attention and salience 知觉的精确性、注意力和显著性

$\displaystyle{ \dfrac{\partial E^{total}(Y^{VP},X^{SN},x^{CN},y^{KN})}{\partial y^{SN}_{mn}}=x^{CN}_{mn}-b^{CN}\varepsilon^{CN}_{nm}+b^{CN}\sum_{k}(\varepsilon^{KN}_{knm}) }$

## Active inference 主动推理

### Active inference and cognitive neuroscience 主动推理与认知神经科学

Category:Biological systems

Category:Systems theory

Category:Computational neuroscience

Category:Mathematical and theoretical biology

This page was moved from wikipedia:en:Free energy principle. Its edit history can be viewed at 自由能原理/edithistory

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