# A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Present Article

**Illustration**

**1.**

#### 1.2. Related Work

## 2. Possible Worlds, Propositions, and Discernment

## 3. Probabilities

#### 3.1. Degrees of Beliefs and Sigma Algebras

**Remark**

**1.**

#### 3.2. Bayes’ Rule and Posterior Probabilities

**Remark**

**2.**

**Remark**

**3.**

**Illustration**

**2**

**Illustration**

**3.**

#### 3.3. Expected Posterior Beliefs

## 4. Learning

#### 4.1. Active Information for Quantifying the Amount of Learning

**Definition**

**1**

**Remark**

**4.**

**Remark**

**5.**

#### 4.2. Learning as Hypothesis Testing

#### 4.3. The Bayesian Approach to Learning

#### 4.4. Test Statistic When ${x}_{0}$ Is Unknown

## 5. Knowledge Acquisition

#### 5.1. Knowledge Acquisition Goes beyond Learning

#### 5.2. A Formal Definition of Knowledge Acquisition

- If $\mathcal{X}\subset {\mathbb{R}}^{q}$, we use the Euclidean distance $d({x}_{1},{x}_{2})=\sqrt{{\sum}_{i=1}^{q}{({x}_{2i}-{x}_{1i})}^{2}}$ between ${x}_{1},{x}_{2}\in \mathcal{X}$ as metric.
- If $\mathcal{X}={\{0,1\}}^{q}$ consists of all binary sequences of length q, then $d({x}_{1},{x}_{2})={\sum}_{i=1}^{q}|{x}_{2i}-{x}_{1i}|$ is the Hamming distance between ${x}_{1}$ and ${x}_{2}$.
- If $\mathcal{X}$ is a finite categorical space, we put$$d({x}_{1},{x}_{2})=\left\{\begin{array}{cc}0,\hfill & {x}_{1}={x}_{2},\hfill \\ 1,\hfill & {x}_{1}\ne {x}_{2}.\hfill \end{array}\right.$$

**Definition**

**2**

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Illustration**

**4**

## 6. Learning and Knowledge Acquisition Processes

#### 6.1. The Process of Discernment and Data Collection

#### 6.2. Strong Learning and Knowledge Acquisition

**Definition**

**3**

**Definition**

**4**

**Illustration**

**5**

#### 6.3. Weak Learning and Knowledge Acquisition

**Definition**

**5**

**Definition**

**6**

## 7. Asymptotics

#### 7.1. Asymptotic Learning and Knowledge Acquisition

**Definition**

**7**

**Definition**

**8**

#### 7.2. Bayesian Asymptotic Theory

**Proposition**

**1.**

**Remark**

**10.**

**Remark**

**11.**

**Theorem**

**1.**

**Proof.**

## 8. Examples

**Example**

**1**

**Example**

**2**

**Example**

**3**

**Example**

**4**

**Example**

**5**

## 9. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of the density function $y\to {f}_{P\left(A\right)}\left(y\right)$ of $P\left(A\right)$ when the data set $D\sim L(\xb7|{x}_{0})$ varies according to the likelihood of the true world parameter for two scenarios where S is either false (

**a**) or true (

**b**). The threshold of the hypothesis test (19) is ${I}^{+}=log[p/{P}_{0}\left(A\right)]$, so that ${H}_{0}$ is rejected when $P\left(A\right)\le p=0.5$. Note that $\overline{P}\left(A\right)$ is the expected value of each density, whereas the error probabilities of type I and II correspond to the areas under the curves in (

**b**) and (

**a**) to the left and right of p, respectively.

**Figure 2.**Illustration of the difference between learning and knowledge acquisition for a scenario with a set of worlds $\mathcal{X}=[0,1]$ and a statement S whose truth function $x\to f\left(x\right)$ is depicted to the left (

**a**) and right (

**b**). It is assumed that S is true (${x}_{0}\in A$), and that the degrees of beliefs ${P}_{0}$ of an ignorant person correspond to a uniform distribution on $\mathcal{X}$. The filled histograms correspond to the density functions $p\left(x\right)dx=dP\left(dx\right)$ of two agent’s beliefs. The agent to the left (

**a**) has learnt about S but not acquired knowledge, since ${x}_{0}$ does not belong to the support of P. The agent to the right has not only learnt about S, but also acquired knowledge, since his belief is justified, corresponding to a distribution P that is more concentrated around the true world ${x}_{0}$, compared to the ignorant person. Hence, the JTB condition is satisfied for the agent to the right, but not for the agent to the left.

**Figure 3.**Degree of belief is represented as a function of coin tosses. There is no strong learning because the belief oscillates. However, there is weak learning after a few coin tosses. In particular, when the number of coin tosses is 1000, there is weak learning since ${P}_{1000}\left(A\right)>{P}_{0}\left(A\right)$ and ${I}_{1000}^{+}\left(A\right)>0$.

**Figure 4.**Posterior densities ${p}_{1}\left(x\right)$ and ${p}_{2}\left(x\right)$ after one and two steps of the discerment and data collection process of Example 2 when S is true (${x}_{0}\in [a,b]$). Since ${p}_{1}$ is measurable with respect to ${\mathcal{G}}_{1}$, it is piecewise-constant with step length $\delta $. Note that strong learning and knowledge acquisition occurs.

**Figure 5.**Posterior densities ${p}_{1}({x}_{1},{x}_{2})$ and ${p}_{2}({x}_{1},{x}_{2})$ after one and two steps of data collection for Example 3. Since ${x}_{01}<a$, it is not possible to have outdoor activities the first day of the camp. The weather forecast density ${p}_{1}$ is supported and piecewise-constant on the four rectangles with width ${\delta}_{1}$ and height ${\delta}_{2}$, corresponding to $\sigma $-algebra ${\mathcal{G}}_{1}$. The true temperatures $({x}_{01},{x}_{02})$ are within the support of ${p}_{1}$. On the other hand, the prophet incorrectly predicts that outdoor activities are possible both days; ${p}_{2}$ is supported on the ellipse. In this case, neither (weak or strong) learning nor knowledge acquisition takes place.

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**MDPI and ACS Style**

Hössjer, O.; Díaz-Pachón, D.A.; Rao, J.S. A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning. *Entropy* **2022**, *24*, 1469.
https://doi.org/10.3390/e24101469

**AMA Style**

Hössjer O, Díaz-Pachón DA, Rao JS. A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning. *Entropy*. 2022; 24(10):1469.
https://doi.org/10.3390/e24101469

**Chicago/Turabian Style**

Hössjer, Ola, Daniel Andrés Díaz-Pachón, and J. Sunil Rao. 2022. "A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning" *Entropy* 24, no. 10: 1469.
https://doi.org/10.3390/e24101469