How different layers of meaning in art interact and contribute to the overall interpretation, accounting for the inherent uncertainty and complexity in our understanding of the world.
Karl Friston’s Free Energy Principle
- The relation to human cognitive theories.
- Potential as filling the gap from AI to AGI.
Diving into the Bayesian Structural Model
- George Box principle about models
- The statistical formula of the bell curve
- The spectrum nature of the beliefs
- The hierarchical nature of the beliefs
- Contemporary art as a sample for hierarchical abstraction of beliefs
Karl Friston’s Free Energy Principle
Karl Friston’s Free Energy Principle offers an intriguing explanation for our intrinsic epistemic craving to continuously learn and understand the world around us. This principle posits that our survival instinct drives us to create mental maps of our environment to minimize surprises and better cope with external changes. Essentially, it suggests that we are wired to seek knowledge to reduce uncertainty and predict future states effectively.
The Free Energy Principle is a theory about cognitive behavior grounded in the observation that while we are constantly bombarded with information from the world, our cognitive capacity to process and internalize this input is exceedingly limited compared to the vast amount of information and its perpetual changes in the universe. In other words, it is impossible for our minds to perfectly model the entire universe. Instead, we can only model what we can perceive and comprehend, using these models to make educated guesses about the rest.
This theory underscores the importance of learning and adaptation in human cognition, as it highlights our need to continuously update our mental models based on new information to maintain a sense of order and predictability in an inherently unpredictable world.
Understand free energy intuitively
https://awjuliani.medium.com/a-gentle-introduction-to-the-free-energy-principle-03f219853177
Diving into the Bayesian Structural Model
George Box principle about models
Humans are essentially guessing machines. We do not have a deterministic, perfect understanding of the world around us. Instead, our comprehension consists of educated guesses organized in a hierarchical structure. This structure is more accurately described as a hierarchical collection of a spectrum of beliefs.
Why a spectrum? This is where the Bayesian component comes into play. Bayesian inference allows us to update our beliefs based on new evidence, continuously refining our mental models. This process involves integrating prior knowledge with incoming data to calculate the probabilities of different hypotheses. Consequently, our beliefs are not static but exist on a spectrum, reflecting varying degrees of certainty and probability.
Through this Bayesian framework, our cognitive system maintains a dynamic, probabilistic understanding of the world. It continuously adjusts and calibrates our beliefs to align with the most plausible explanations given the available information. Thus, we navigate and make sense of an ever-changing world by constantly refining our educated guesses.
Diving into the Bayesian Structural Model
Spectrum of beliefs
Imagine the Gaussian distribution as a representation of the spectrum of your beliefs about a particular subject. For instance, if you consider the weather forecast, the peak of the Gaussian curve (around x=0) might represent your belief that it is most likely to be sunny and clear. However, the tails of the distribution reflect the smaller possibilities that the weather could be rainy and not clear.
This visualization helps to understand that every idea we have about the world is composed of a spectrum of possibilities, distributed from the most likely to the unlikely. Our beliefs are not confined to single, certain outcomes but instead cover a range of possibilities, with some scenarios being more probable than others. This probabilistic nature of our beliefs allows us to account for uncertainty and adjust our expectations based on new information, much like the Bayesian approach to updating beliefs.
Bayesian Statistics
Bayesian statistics is highly regarded in practice due to its ability to incorporate prior information, provide intuitive interpretations of uncertainty, enhance predictive performance, and offer flexible modeling capabilities. These attributes make it a powerful tool for a wide range of practical applications.
Bayesian Statistics Advantages
Bayesian statistics has several attributes that make it particularly advantageous for practical applications:
- Uncertainty Quantification:
Bayesian methods provide a comprehensive framework for quantifying uncertainty. By generating posterior distributions, they offer a complete picture of the uncertainty surrounding parameter estimates and predictions. - Decision-Theoretic Approach:
Bayesian statistics aligns well with decision theory, offering a structured way to make decisions under uncertainty. The posterior distribution can be used to calculate expected utilities or losses, aiding in optimal decision-making. - Hierarchical Modeling:
Bayesian statistics excels in hierarchical modeling, which is useful for analyzing data with multiple levels of variability (e.g., multi-level or mixed-effects models). This is particularly valuable in fields like healthcare, social sciences, and ecology. - Handling of Missing Data:
Bayesian methods can naturally handle missing data by treating missing values as additional parameters to be estimated. This approach allows for more accurate and robust analyses compared to traditional imputation methods. - Computational Advances:
Advances in computational techniques, such as Markov Chain Monte Carlo (MCMC) methods and variational inference, have made Bayesian methods more accessible and feasible for complex, high-dimensional problems.
Breaking down the formula
The formula shown in the image is Bayes’ Rule, a fundamental theorem in Bayesian statistics. Here’s a detailed explanation of each component of the
Let’s break it down:
1. P(hypothesis | data):
This is the posterior probability. It represents the probability of the hypothesis being true given the observed data. In Bayesian inference, this is what we aim to compute.
2. P(data | hypothesis):
This is the likelihood. It represents the probability of observing the data given that the hypothesis is true. It shows how likely the observed data is under the hypothesis.
3. P(hypothesis):
This is the prior probability. It represents the initial probability of the hypothesis being true before observing the data. This incorporates any existing knowledge or beliefs about the hypothesis.
4. P(data):
This is the marginal likelihood or evidence. It represents the total probability of observing the data under all possible hypotheses. It can be calculated by summing or integrating the likelihood across all possible hypotheses, weighted by their prior probabilities.
The formula can be understood as a way to update our beliefs about the hypothesis after observing the data:
The prior probability (P(hypothesis)) is updated with the likelihood (P(data | hypothesis)) to form the posterior probability (P(hypothesis | data)).
The evidence (P(data)) serves as a normalizing constant ensuring that the sum of the posterior probabilities over all possible hypotheses equals 1.
In summary, Bayes’ Rule provides a mathematical framework for updating the probability of a hypothesis based on new evidence, combining prior knowledge with observed data to produce a revised probability.
Bayesian Inference
Bayesian Inference is the specific process within Bayesian statistics that applies Bayes’ Theorem to update beliefs and make statistical inferences.
Bayesian inference techniques specify how one should update one’s beliefs upon observing data.
The hierarchical structure of believes
Our beliefs are organized hierarchically, where each specific belief is related to more general or abstract beliefs above it. For example, consider the belief that “In Africa, it has been getting warmer over the last few years.” This specific belief is part of a broader belief that encompasses similar observations in different regions:
In Africa, it has been getting warmer over the last few years.
The same trend is observed in Europe.
Similar warming trends are noted in other parts of the world.
At a higher level, these regional beliefs coalesce into a more general belief: “The whole world is getting warmer.”
This hierarchical structure allows us to integrate detailed, localized observations into more abstract, overarching conclusions. It enables our cognitive system to manage complexity by organizing information into nested layers, each layer summarizing and abstracting the information from the levels below. This approach helps us to form coherent, comprehensive understandings of broad phenomena based on specific, local observations.
Contemporary art as an example
As a radical example, lets take ‘Fragmented Reflections’ image at the beginning of this article.
Description: “Fragmented Reflections” is a mixed media artwork that intricately explores the complexity of human identity and the multifaceted nature of existence. At first glance, the piece presents a fragmented human face composed of a collage of different materials, textures, and colors. Each fragment represents a different aspect of the human experience, from joy and love to sorrow and despair.
The background is a juxtaposition of urban and natural elements, with skyscrapers and trees blending seamlessly into each other. This symbolizes the ongoing tension between technological advancement and the natural world, urging viewers to contemplate their place within this dichotomy.
Some parts of the face are transparent, revealing an underlayer of complex patterns and abstract symbols. These hidden elements invite viewers to delve deeper, suggesting that there is always more beneath the surface of what we see. The abstract symbols interspersed throughout the piece represent various cultural and personal symbols, contributing to a sense of universal interconnectedness.
The vibrant colors evoke a range of emotions, while the intricate textures invite viewers to experience the artwork both visually and tactilely. The overall composition is both harmonious and disjointed, reflecting the fragmented yet interconnected nature of human identity and experience.
This piece serves as a metaphor for the complexities of modern life, encouraging viewers to reflect on their own identities and the broader societal forces at play.
Level 1: Basic Elements (Observed Data)
- Fragmented Human Face: Represents basic visual components.
- Textures (Ti): Various surface qualities.
- Colors (Cj): Different hues and shades.
- Shapes (Sk): Geometric forms and outlines.
Level 2: Symbolic Representation (Latent Variables)
- Emotional States (El): Abstract emotional attributes conveyed by visual elements.
- P(El | Ti, Cj, Sk): Probability distribution over emotional states given textures, colors, and shapes.
Level 3: Contextual Background (Latent Variables)
- Urban-Nature Contrast: Tension between urban and natural elements.
- Urban elements (U)
- Natural elements (N)
- P(U, N | El): Probability distribution over urban and natural elements given emotional states.
Level 4: Deeper Meanings (Latent Variables)
- Symbolic Abstraction (Am): Deeper, abstract meanings such as cultural symbols and personal reflections.
- P(Am | U, N): Probability distribution over abstract meanings given the urban and natural contrast.
Level 5: Holistic Interpretation (Final Layer)
- Overall Interpretation (I): The final interpretation of the artwork as a whole.
- P(I | Am, El): Probability distribution over overall interpretations given abstract meanings and emotional states.
Interpretation:
- The model starts with the observed visual elements (textures, colors, shapes).
- These elements probabilistically influence the perception of emotional states.
- Emotional states, in turn, influence the perception of urban and natural elements, representing the contextual background.
- The contextual background informs deeper symbolic abstractions.
- Finally, both the symbolic abstractions and emotional states combine to form the overall interpretation of the artwork.
This Bayesian structured model allows us to understand how different layers of meaning in the artwork interact and contribute to the overall interpretation, accounting for the inherent uncertainty and complexity in interpreting contemporary art.
