Finding emergence in data by maximizing effective information

The climate system, ecosystems, bird flocks, ant colonies, cells, brains, and many other complex systems are composed of numerous interacting elements and exhibit a wide range of nonlinear dynamical behaviors [1, 2]. In the past few decades, the research topic of data-driven modeling complex systems has gained significant attention, driven by the increasing availability and accumulation of data from real dynamical systems [3, 4, 5, 6]. However, complex systems always exhibit emergent behaviors [7, 1]. That means some interesting emergent patterns or dynamical behaviors such as waves [8], periodic oscillations [9] and solitons [10] can hardly be directly observed and identified from the micro-level behavioral data. Therefore, the identification and measure of emergence and the capture of emergent dynamical patterns solely from observational raw data have become crucial challenges in complex systems research [11, 12, 13, 14]. But in order to address these problems, it is necessary to first develop a quantitative understanding of emergence.

Emergence, as a distinctive feature of complex systems [7], has historically been challenging to quantify and describe in quantitative terms [15, 16, 17, 18]. Most conventional measures or methods either rely on predefined macro-variables (e.g., [19, 20, 21]) or are tailored to specific scenarios in engineered systems (e.g. [17, 22, 23, 24]). However, there is a need for a unified method to quantify emergence across different contexts. The theory of causal emergence (CE), introduced by Erik Hoel in 2013, offers a framework to tackle this challenge [25, 26]. Hoel’s theory aims to understand emergence through the lens of causality. The connection between emergence and causality is implied in the descriptive definition of emergence, as stated in the work by J. Fromm[27]. According to this definition, a macro-level property, such as patterns or dynamical behaviors, is considered emergent if it cannot be explained or directly attributed to the individuals in the system. The theory of causal emergence formalizes this concept within the framework of discrete Markov dynamical systems. As shown in Figure 1(a), Hoel et al’s theory states that if a system exhibits stronger causal effects after a specific coarse-graining transformation compared to the original system, then CE has occurred. In Hoel’s framework, the degree of causal effect in a dynamical system is quantified using effective information (EI) [28]. EI can be understood as an intervention-based version of mutual information between two successive states in a dynamical system over time. It is a measure that solely depends on the system’s dynamics. If a dynamical system is more deterministic and non-degenerate, meaning that the temporal adjacent states can be inferred from each other in both directions of the time arrow, then it will have a larger EI [25]. This measure has been shown to be compatible with other well-known measures of causal effect [29]. Figure 1(b) gives an example of CE for simple Markov chain.

While the causal emergence theory has successfully quantified emergence using EI and has found applications in various fields [30, 31, 32], there are some drawbacks. Firstly, the markov transition matrix of the micro-dynamics should be given rather than constructed from data. Secondly, a predefined coarse-graining strategy must be provided or optimized through maximizing EI, but this optimization process is computationally complex[33, 30]. Although Rosas et al.[11] proposed a new framework for CE based on partial information decomposition theory [34], which does not require a predefined coarse-graining strategy, it still involves iterating through all variable combinations on the information lattice to compare synergistic information, resulting in significant computational complexity. Rosas et al. also proposed an approximate method to mitigate the complexity [11], but it requires a pre-defined macro-variable.

Therefore, the challenge of finding emergence in data, which involves determining whether CE has occurred within a system and to what extent based solely on observational data of its behavior, remains unresolved. The most daunting task is that all the elements, including the markov dynamics at both the micro- and macro-levels, as well as the coarse-graining strategy to obtain macro-variables, need to be learned from raw data and cannot be predefined in advance [12]. Once the learning process is completed, we can compare the strength of causal effects (measured by EI) in dynamics at different scales to identify CE from the data. Therefore, the problem of finding CE within Hoel’s theoretical framework is essentially equivalent to the challenge of data-driven modeling within a coarse-grained space for complex systems [12]. Building models for complex systems at multiple coarse-grained levels within learned emergent spaces is of utmost importance for both identifying CE and conducting data-driven modeling in complex systems.

Recently, several machine learning frameworks have emerged for learning and simulating the dynamics of complex systems within coarse-grained latent or hidden spaces [35, 14, 36, 37, 38]. By learning the dynamics in a hidden space, these frameworks enable the removal of noise and irrelevant factors from raw data, enhancing prediction capabilities and adaptability to diverse environments. While these learning systems can capture emergent dynamics, they may not directly address the fundamental nature of CE, which entails stronger causality. According to Judea Pearl’s hierarchy of causality, prediction-based learning is situated at the level of association and cannot address the challenges related to intervention and counterfactuals [39]. Empirically, dynamics learned solely based on predictions may be influenced by the distributions of the input data, which can be limited by data diversity and the problem of over fitting models [40]. However, what we truly desire is an invariant causal mechanism or dynamics that are independent of the input data. This allows the learned mechanism or dynamics to be adaptable to broader domains, generalizable to areas beyond the distribution of training data, and capable of accommodating diverse interventions [41, 42, 43]. Unfortunately, few studies have explored the integration of causality and latent space dynamics to address the challenges of data-driven modeling in complex systems [44].

Inspired by the theory of causal emergence, this paper aims to address the challenge of learning causal mechanisms within a learned coarse-grained macro-level(latent) space. The approach involves maximizing the EI of the emergent macro-level dynamics, which is equivalent to maximizing the degree of causal effect in the learned coarse-grained dynamics [29]. By artificial intervening the distributions of input data as uniform distribution, this maximization process helps separating the invariant causal mechanisms from the variant data distribution as much as possible. To achieve this, a novel machine learning framework called Neural Information Squeezer Plus (NIS+) is proposed. NIS+ extends the previous framework (NIS) to solve the problem of maximizing EI under coarse-grained representations. As shown in Figure 2, NIS+ not only learns emergent macro-dynamics and coarse-grained strategies but also quantifies the degree of CE from time series data and captures emergent patterns. Mathematical theorems ensure the flexibility of our framework in different application scenarios, such as cellular automata and multi-agent systems. Numerical experiments demonstrate the effectiveness of NIS+ in capturing emergent behaviors and finding CE in various simulations, including SIR dynamics[45], bird flocking (Boids model)[46], and the Game of Life[47]. Additionally, NIS+ is applied to find emergent features in real brain data from 830 subjects while watching the same movie. Experiments also demonstrate that the learned dynamical model exhibits improved performance in generalization compared to other models.

Finding CE in time series data involves two sub-problems: emergent dynamics learning and causal emergence quantification. In the first problem, both the dynamics in micro- and macro-level and the coarse-graining strategy should be learned from data. After that, we can measure the degree of CE and judge if emergence occurs in data according to the learned micro- and macro-dynamics.

We will validate the effectiveness of the NIS+ framework through numerical experiments where data is generated by different artificial models (dynamical systems, multi-agent systems, and cellular automata). Additionally, we will apply NIS+ to real fMRI data from human subjects to uncover interesting macro-level variables and dynamics. In these experiments, we will evaluate the models’ prediction and generalization abilities. We will also assess their capability to identify CE and compare it with Rosas’ $\Psi$ indicator proposed in [11], an alternative measure for quantifying CE approximately. For the experiments, a threshold of normalized MAE of 0.3 was selected based on the observation that exceeding this value leads to a significant change in the relationship between normalized MAE and noise (see below).

Inspired by the theory of causal emergence, this paper introduces a novel machine learning framework called Neural Information Squeezer Plus (NIS+) to learn emergent macro-dynamics, and suitable coarse-graining methods directly from data. Additionally, it aims to quantify the degree of CE under various conditions.

The distinguishing feature of our framework, compared to other machine learning frameworks, is its focus on maximizing the effective information (EI) of the learned macro-dynamics while maintaining effectiveness constraints. This enables the learned emergent macro-dynamics to capture the invariant causal mechanism that is as independent as possible from the distribution of input data. This feature not only enables NIS+ to identify CE in data across different environments but also enhances its ability for generalization on the environments which are distinct from training data. By introducing the error constraint in Equation LABEL:old_optimization, we can address the issues raised in [48] and enhance the robustness of the maximization of EI framework. As a result, NIS+ extends Hoel’s theory of CE to be applicable to both discrete and continuous dynamical systems, as well as real data.

Three experiments were conducted to evaluate the capabilities of NIS+ in learning and generalization, and quantifying CE directly from data. Furthermore, we applied this framework to the domain of the Game of Life (refer to the support information section 13). These experiments encompassed three simulation scenarios and one real fMRI dataset for 830 human subjects while watching same movie clips.

The experiments indicate that by maximizing EI, NIS+ outperforms other machine learning models in tasks such as multi-step predictions and pattern capturing, even in environments that were not encountered during the training process. Consequently, NIS+ enables the acquisition of a more robust macro-dynamics in the latent space.

Furthermore, the experiments show that NIS+ can quantify CE in a more reasonable way than Rosas’ $\Psi$ indicator. With this framework, we can distinguish different scenarios in the data and identify which settings contain more regular patterns, as demonstrated in the experiment conducted on the Game of Life. The experiment on the Boid model also provides insights into how two types of noise can impact the degrees of CE. The conclusion is that extrinsic noise may increase CE, while intrinsic noise may decrease it. This indicates that extrinsic noise, arising from observational uncertainty, can be mitigated by the learned coarse-graining strategy. On the other hand, intrinsic noise, stemming from inherent uncertainty in the dynamical rules, cannot be eliminated.

The process of coarse-graining a complex system can be effectively learned by NIS+ and illustrated using the Integrated Gradients (IG) method, as demonstrated in the experiments involving the Boid model and the brain dataset. Through this method, the relationship between macro-states and micro-states can be visualized, allowing for the identification of the most significant variables within the micro-states. In the brain experiment, it was observed that the most crucial information stored in the unique macro-state exhibited a strong correlation with the micro-variables of the visual subnetwork. This finding highlights the ability of NIS+ to identify an appropriate coarse-graining strategy that aligns with meaningful biological interpretations.

NIS+ holds potential for various applications in data-driven modeling of real complex systems, such as climate systems, collective behaviors, fluid dynamics, brain activities, and traffic flows. By learning more robust macro-dynamics, the predictive capabilities of these systems can be enhanced. For instance, El Niño, which arises from the intricate interplay of oceanic and atmospheric conditions, exemplifies the emergence of a major climatic pattern from underlying factors. Understanding these emergent macro-dynamics can be instrumental in modeling and predicting El Niño events. By leveraging NIS+ to capture and quantify the CE in such complex systems, we can gain valuable insights and improve our ability to forecast their behavior.

Further extensions of the EI maximization method to other problem domains, such as image classification and natural language understanding, in addition to dynamics learning, warrant attention. Specifically, applying this principle to reinforcement learning agents equipped with world models has the potential to enhance performance across various environments. By incorporating the EI maximization approach into these domains, we can potentially improve the ability of models to understand and interpret complex visual and textual information. This can lead to advancements in tasks such as image recognition, object detection, language understanding, and machine translation.

The relationship between causal emergence and causal representation learning (CRL)[41] deserves further exploration. The NIS+ framework can be seen as a form of CRL, where the macro-dynamics serve as the causal mechanism and the coarse-graining strategy acts as the representation. As such, techniques employed in CRL can be applied to find CE in data. Conversely, the concept of CE and coarse-graining can be leveraged in CRL to enhance the interpretability of the model. By integrating the principles and methodologies from both CE and CRL, we can potentially develop more powerful and interpretable models. This synergy can lead to a deeper understanding of the causal relationships within complex systems and enable the extraction of meaningful representations that capture the emergent patterns and causal mechanisms present in the data. Further research in this direction can contribute to advancements in both CE and causal representation learning.

Another interesting merit of NIS+ is its potential contribution to emergence theory by reconciling the debate on whether emergence is an objective concept or an epistemic notion dependent on the observer. By designing a machine to maximize EI, we can extract objective emergent features and dynamics. The machine serves as an observer, but an objective one. Therefore, if the machine observer detects interesting patterns in the data, emergence occurs.

However, there are several limitations in this paper that should be addressed in future studies. Firstly, the requirement of a large amount of training data for NIS+ to learn the macro-dynamics and coarse-graining strategy may not be feasible in many real-world cases. If the training is insufficient, it may lead to incorrect identification of CE. Therefore, it is necessary to incorporate other numeric method, such as Rosas’ $\Phi$ID [11], to make accurate judgments. One advantage of NIS+ is its ability to identify coarse-grained macro-states, which can then be used as input for Rosas’ method. Secondly, the interpretability of neural networks, particularly for the macro-dynamics learner, remains a challenge. Enhancing the interpretability of the learned models can provide valuable insights into the underlying mechanisms and improve the trustworthiness of the results. Additionally, the learned macro-dynamics in this work are homogeneous, resulting in the same EI value across different locations in the macro-state space. However, in real-world applications, this may not always hold true. Therefore, it is necessary to develop local measures for CE, especially for systems with heterogeneous dynamics. This would facilitate the identification of local emergent patterns, making the analysis more nuanced and insightful. Lastly, the current framework is primarily designed for Markov dynamics, while many real complex systems exhibit long-term memory or involve unobservable variables. Extending the framework to accommodate non-Markov dynamics is an important area for future research. Addressing these limitations and exploring these avenues for improvement will contribute to the advancement of the field and enable the application of NIS+ to a wider range of complex systems.

In order to provide a comprehensive understanding of the article, we will first dedicate a certain amount of space to introduce Erik Hoel’s theory of causal emergence, and we introduce the details of the NIS and NIS+ frameworks and other used techniques. After that, the details of the fMRI time series data is introduced.

The authors would like to acknowledge all the members who participated in the "Causal Emergence Reading Group" organized by the Swarma Research and the "Swarma - Kaifeng" Workshop. We are especially grateful to Professors Yizhuang You, Peng Cui, Jing He, Dr.Chaochao Lu, and Dr. Yanbo Zhang for their invaluable contributions and insightful discussions throughout the research process. We also thank the large language model ChatGPT for its assistance in refining our paper writing.

Support Information

The basic idea of Hoel’s theory is that we have two different views (micro and macro) of a same Markov dynamical system, and CE occurs if the macro-dynamics has stronger causal effect than the micro-dynamics. As shown in Figure 1, for example, the micro-scale system is composed by many colliding balls. The macro-scale system is a coarse-grained description of the colliding balls with colorful boxes where each box represents the number of balls within the box. In Figure 1, the vertical axis represents the scale(micro- and macro-), and the horizontal axis represents time, which depicts the evolution of the system’s dynamics in both scales. All the dynamical systems considered are Markovian, therefore two temporal successive snapshots are necessary. The dynamics in both micro- and macro-scales are represented by transitional probability matrix(TPM): $f_{m}$ for micro-dynamics and $f_{M}$ for macro-dynamics.

We can measure the strength of its causal effects for each dynamics(TPM) by effective information (EI). EI can be understood as an intervention-based version of mutual information between two successive states in a dynamical system over time. For any TPM $f$, EI can be calculated by:

where, $S_{t}$ is the random variable to represent the state of the system at time $t$, $do(S_{t-1}\sim U(S))$ represents the do-operator [39] that intervene the state of the system at time $t-1$ to force that $S_{t-1}$ follows a uniform distribution on the value space $S$ of $S_{t}$. $f_{ij}$ is the probability at the $i$th row and the $j$th column. $N=|S|$ is the total number of states and is also the number of rows or columns. $\log$ represents the logarithmic operation with a base of 2. Equation 1 can be further decomposed into two terms: determinism and non-degeneracy which characterize that how the future state can be determined by the past state and vice versa, respectively[25].

With EI, we can compare two TPMs. If $\mathcal{J}(f_{M})$ is larger than $\mathcal{J}(f_{m})$, we say CE occurs in this dynamical system. In practice, we calculate another value called causal emergence to judge if CE takes place, that is:

Therefore, if $\Delta\mathcal{J}>0$, then CE occurs. $\Delta\mathcal{J}$ can be treated as the quantitative measure of emergence for Markov dynamics. An example of CE on Markov chain is shown in Figure 1(b).

The measure of EI of discrete Markov chains can be calculated by Equation 1, however, it is not suitable for neural networks because a neural network is a deterministic function. Therefore, we need to extend the definition of EI for general neural networks. We continue to use the method defined in [12]. The basic idea is to understand a neural network as a Gaussian distribution conditional on the input vector with the mean as the deterministic function of the input values, and the standard deviations as the mean square errors of this neural network on the training or test datasets.

In general, if the input of a neural network is $X=(x_{1},x_{2},\cdot\cdot\cdot,x_{n})\in[-L,L]^{n}$, which means $X$ is defined on a hyper-cube with size $L$, where $L$ is a very large integer. The output is $Y=(y_{1},y_{2},\cdot\cdot\cdot,y_{m})$, and $Y=\mu(X)$. Here $\mu$ is the deterministic mapping implemented by the neural network: $\mu:\mathcal{R}^{n}\rightarrow\mathcal{R}^{m}$, and its Jacobian matrix at $X$ is $\partial_{X^{\prime}}\mu(X)\equiv\left\{\frac{\partial\mu_{i}(X^{\prime})}{\partial X^{\prime}_{j}}\left|{}_{X^{\prime}=X}\right.\right\}_{nm}$. If the neural network can be regarded as a Gaussian distribution conditional on given $X$, then the effective information (EI) of the neural network can be calculated in the following way:

where, $\Sigma=diag(\sigma_{1}^{2},\sigma_{2}^{2},\cdot\cdot\cdot,\sigma_{m}^{2})$ is the co-variance matrix, and $\sigma_{i}$ is the standard deviation of the output $y_{i}$ which can be estimated by the mean square error of $y_{i}$ under given $x_{i}$. $U([-L,L]^{n})$ is the uniform distribution on $[-L,L]^{n}$, and $|\cdot|$ is absolute value, and $\det$ is determinant. If $\det(\partial_{X^{\prime}}\mu(X))\equiv 0$ for all $X$, then we set $EI\approx 0$.

However, Equation 3 can not be applied in real cases directly because it will increase as the dimension of input $n$ or output $m$ increases[12]. Therefore, larger EI is always expected for the models with higher dimension. The way to solve this problem is by dividing the input dimension to define dimension averaged effective information(dEI) and is denoted as $\mathcal{J}$:

When the numbers of input and output are identical ($m=n$, this condition is always hold for all the results reported in the main text), Equation 3 becomes:

We always use this measure to quantify EI for neural networks in the main text. However, this measure is also not perfect because it is dependent on a free parameter $L$, the range of the domain for the input data. Our solution is to calculate the dimension averaged CE to eliminate the influence of $L$. For macro-dynamics $f_{M}$ with dimension $q$ and micro-dynamics $f_{m}$ with dimension $p$, we define dimension averaged CE as:

If $f_{M}$ and $f_{m}$ are parameterized by neural networks of $\mu_{M}$ with dimension $q$ and $\mu_{m}$ with dimension $p$, then

where $\sigma_{i,M}$ and $\sigma_{i,m}$ are the standard deviations for $\mu_{M}$ and $\mu_{m}$ on the $i$th dimension, respectively. At this point, the influences of the input or output dimensions and the parameter $L$ has been completely eliminated, making it a more reliable indicator. Therefore, the comparisons between the learned macro-dynamics reported in the main text are reliable because the measure of dimension averaged CE is used.

In practice, the first step is to select a sufficiently large value for $L$ to ensure that all macro-states are encompassed within the region $[-L,L]^{q}$. Subsequently, the determinant of Jacobian matrices can be estimated using Monte Carlo integration, which involves sampling data points within this region. The standard deviations can be estimated using a test dataset, and the probability reweighting technique is employed to account for interventions and ensure a uniform distribution.

In NIS, both the encoder($\phi$) and the decoder($\phi^{{\dagger}}$) use the invertible neural network module RealNVP[56].

Concretely, if the input of the module is $\mathbf{x}$ with dimension $p$ and the output is $\mathbf{x^{\prime}}$ with the same dimension, then the RealNVP module can perform the following computation steps:

where, $m$ is an integer in between 1 and $p$.

where, $\bigotimes$ is element-wised time, $s_{1},s_{2},t_{1}$ and $t_{2}$ are feed-forward neural networks with arbitrary architectures, while their input-output dimensions must match with the data. For the experiments in this article, they each have two intermediate hidden layers. The input and output layers have the same number of neurons as the dimensions of the micro-state samples. Each hidden layer has 64 neurons, and the output of each hidden layer is transformed by the non-linear activation function LeakyReLU. In practice, $s_{1}$ or $s_{2}$ always do an exponential operation on the output of the feed-forward neural network to facilitate the inverse computation.

Finally,

It is not difficult to verify that all three steps are invertible. Equation 9 is invertible because the same form but with negative signs can be obtained by solving the expressions of $\mathbf{x_{1}}$ and $\mathbf{x_{2}}$ with $\mathbf{x^{\prime}_{1}}$ and $\mathbf{x^{\prime}_{2}}$ from Equation 9.

To simulate more complex invertible functions, we always duplex the basic RealNVP modules by stacking them together. In the main text, we use duplex the basic RealNVP module by three times.

Due to the reversibility of RealNVP, if its output is used as input in a subsequent iteration, the new output will be the same as the original input. Therefore, the neural networks used in the encoder-decoder can share parameters. The process of encoding to obtain the macro-state involves projecting the forward output and retaining only the first few dimensions. On the other hand, the process of decoding to obtain the micro-state involves concatenating the macro-state with a normal distribution noise, expanding it to match the micro-state dimensions, and then feeding it back into the entire invertible neural network.

In order to use inverse probability weighting technique, we need to estimate the probability distribution of the samples. KDE(Kernel Density Estimation) is a commonly used estimation method that can effectively eliminate the influence of outliers on the overall probability distribution estimation. In this experiment, we estimate the probability distribution of macro-states $\boldsymbol{y}_{t}$. Below is an introduction to the operation details of KDE. For a sample $(x_{1},x_{2},\cdot\cdot\cdot,x_{n})$, we have the following kernel density estimation:

$n$ represents the number of samples, the hyperparameter $h$ denotes the bandwidth, which is determined based on the rough range of the data and typically set to 0.05 in this article. $K$ is the kernel, specifically the standard normal density function. After obtaining the estimation function, each sample point is evaluated individually, resulting in the corresponding probability value for each sample point. By dividing the probability of the target distribution (uniform distribution) by our estimated probability value, we obtained the inverse probability weights corresponding to each sample point. To cover all sample points, the range of the uniform distribution needs to be limited by a parameter $L$, which ensures that a square with side length $2L$ can encompass all the sample points in all dimensions.

Another thing to note is that the weights obtained at this point are related to the sample size, so they need to be normalized. However, this will result in very small weight values corresponding to the samples. Therefore, it is necessary to multiply them by the sample quantity to amplify the weight values back to their normal scale. Then, multiply this weight value by the training loss of each sample to enhance the training of sparse regions, achieving the purpose of inverse probability weighting. Since the encoder parameters change with each iteration, causing the distribution of $\boldsymbol{y}_{t}$ to change, we will re-estimate the probability distribution of the entire sample using KDE every 1000 epochs (a total of at least 30000 epochs of iteration, the number of iterations may vary in different experiments).

We present here the computing method for Rosas’ $\Psi$, a measurable indicator of CE proposed by Rosas [11]. Its definition is as follows:

where $Y$ represents the given macrostate, $Y_{t}$ and $Y_{t^{\prime}}$ represent the macrostates at two consecutive time points, and $X_{t}^{j}$ represents the $j$-th dimension of the microstate at time $t$. According to Rosas’ theory, Rosas’ $\Psi$ being greater than 0 is a sufficient but not necessary condition for emergence to occur. In this paper, we employ the method proposed by Lu et al. [57] to compute the continuous variable mutual information in the experimental configuration.

To compare the advantages of the NIS+ framework itself, we constructed a regular fully connected neural network (NN) with 33,668 parameters and a Variational Autoencoder (VAE) with 34,956 parameters in the SIR experiment (the NIS+ model consists of a total of 37,404 parameters, which includes the inverse dynamics component). NN consists of an input layer, an output layer, and four hidden layers with 4, 64, 128, 128, 64, 4 neurons respectively. NN+ refers to the NN that combines inverse probability weighting technique, with the same parameter framework. For each data point $\boldsymbol{x}_{t}$, we estimate the KDE probability distribution and calculate the inverse probability weighting. Finally, the weights are multiplied with the loss to give different training weights to samples with different densities.

VAE is a generative model that maps micro-level data to a normal distribution in latent space. During each prediction, it samples from the normal distribution and then decodes it [58]. It consists of an encoder and a decoder, both of which have 4 hidden layers, each layer containing 64 neurons. The input and predicted output are both four-dimensional variables, and the latent space also has four dimensions, with two dimensions for mean and two dimensions for variance. The macro-dynamic learner fits the dynamical changes of the two-dimensional mean variables. During the training process, the loss function consists of two parts: reconstruction error, which is the MAE error between the predicted output and the true label, and the KL divergence between the normal distribution of the latent space samples and the standard normal distribution. These two parts are combined with a 2:1 ratio and used for gradient backpropagation. VAE+ combines inverse probability weighting and bidirectional dynamic learning in the training of VAE, and adopts the same training method as NIS+. We keep the variance variables obtained from encoding unchanged, and use inverse probability weighting to obtain weight factors for the mean variables. Additionally, a reverse dynamic learner is constructed to predict $\boldsymbol{y}_{t}$ given $\boldsymbol{y}_{t+1}$.

Because NN+ does not have a multi-scale framework for encoding and decoding, it directly estimates and calculates inverse probability weights at the micro-scale data level, and cannot improve the forward dynamics EI through training reverse dynamics. In VAE+, we focus on the dynamics trained in the latent space, aiming to estimate and calculate inverse probability weights in the latent space, and can optimize the encoder by training the reverse dynamics in the latent space to improve EI.The comparison results are shown in Figure 3(b) and (d).

Here we introduce a method called Integrated Gradients[59] that is used to explain the connection between macro-states and micro-states. To formally describe this technique, let us consider a function $F:\mathcal{R}^{p}\to[0,1]$ that represents a deep network. Specifically, let $x\in\mathcal{R}^{p}$ denote the input under consideration, and $x^{\prime}\in\mathcal{R}^{p}$ denote the baseline input. In both the fMRI and Boids experiments, the baseline is set to 0.

We compute the gradients at each point along a straight-line path (in $\mathcal{R}^{p}$) from the baseline $x^{\prime}$ to the input $x$. By accumulating these gradients to obtain the final measure of integrated gradients(IG). More precisely, integrated gradients are defined as the path integral of the gradients along the straight-line path from the baseline $x^{\prime}$ to the input $x$.

The integrated gradient along the i-th dimension for an input x and baseline $x^{\prime}$ is defined as follows. Here, $\partial F(x)/\partial x_{i}$ represents the gradient of $F(x)$ along the $i$-th dimension. The Integrated Gradients(IG) that we have computed are shown in Equation 13.

AOMIC is an fMRI collection which composes AOMIC PIOP1, AOMIC PIOP2 and AOMIC ID1000[53].

The AOMIC PIOP2 collects subjects’ data for mutiple tasks such as emotion-matching, working-memory and so on. Here, we just use 50 subjects’ resting fMRI data since some time steps of other subject’s have been thrown outby removing the effects of artificial motion, global signal, and white matter signal and cerebrospinal fluid signal using fMRIPrep results[60, 61], which means a difficulty in time alignment.

The AOMIC ID 1000 collected when 881 subjects are watching movies. It contains both raw data and preprocsessed data(See more experimental and preprocsessing details in [53]). Here, we should notice that the movie is edited in a way for no clear semantic meanings but just some concatenated images from the movie. Therefore, it is expected that subjects’ brain activation patterns shouldn’t respond to some higher-order functions such as semantic understandings.

When working with fMRI data, the high dimensionality of the data (over 140,000 dimensions) poses computational challenges. To address this, dimension reduction techniques through brain atlas are employed to make the calculations feasible. We first preprocessed the data by removing the effects of artificial motion, global signal, and white matter signal and cerebrospinal fluid signal using fmriPrep results contained in AOMIC ID1000 [60, 61]. Simultaneously, we detrend and normalize the data during this process. By doing the whole preprocessing process, we also throw out 51 subjects’ data since some time steps of these subject’s have been removed. Furthermore, to facilitate the investigation of the correlation between the required brain function for watching movie clips and the emergent macro dynamics, we employed the Schaefer atlas to divide the brain into 100 functional areas. Also, we have committed the philosophy of anatomical Atlas, which seems to suggest the converging evidence(See support information section 14.2). All of these preprocessing steps were performed using Nilearn[62, 63]. During the training process, the only difference is the use of PCA in dimension reduction to handle the KDE approximation required for reweighting if $q>10$.

Before we prove the theorem, let us first introduce the symbols that will be used in the following paragraphs. We will use capital letters to represent the corresponding random variables. For example, $X_{t}$ represents the random variable of the micro-state $\boldsymbol{x}_{t}$ at time $t$, and $Y_{t+1}$ represents the random variable corresponding to the macro-state $\boldsymbol{y}_{t+1}$. For any random variable $X$, $\tilde{X}$ represents the same random variable $X$ after intervention. $\hat{X}$ means the prediction for $X$ given by neural networks.

Second, to understand how the intervention on $Y_{t}$ can affect other variables, a causal graph derived from the framework of NIS and NIS+ and depicts the causal relations among variables is very useful, this is shown in Figure 2. In which, $X_{t}$ and $\hat{X}_{t+1}$ denote the input random variable and predicted output random variable of the NIS+ on a micro scale, while $Y_{t}$ and $\hat{Y}_{t+1}$ represent the input and output of $f_{q}$ on a macro scale. After intervening on $Y_{t}$, $\tilde{X}_{t}$, $\tilde{X}_{t+1},\tilde{Y}_{t},\tilde{Y}_{t+1}$ represent different micro or macro variables under the intervention $do(Y_{t}\sim U(Y))$, corresponding to $X_{t}$,$\hat{X}_{t+1},Y_{t}$, and $\hat{Y}_{t+1}$, respectively.

In this sub-section, we will show the mathematical proves of the theorems mentioned in the main text.