Subjective logic

Key takeaways

Subjective logic belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Subjective logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Subjective logic from memory before moving on to harder problems.

Reference excerpt

Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and relatively unreliable sources. For example, it can be used for modeling and analysing trust networks and Bayesian networks. Arguments in subjective logic are subjective opinions about state variables which can take values from a domain (aka state space), where a state value can be thought of as a proposition which can be true or false. A binomial opinion applies to a binary state variable, and can be represented as a Beta PDF (Probability Density Function). A multinomial opinion applies to a state variable of multiple possible values, and can be represented as a Dirichlet PDF (Probability Density Function). Through the correspondence between opinions and Beta/Dirichlet distributions, subjective logic provides an algebra for these functions. Opinions are also related to the belief representation in Dempster–Shafer belief theory. A philosophical aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false, because our imperfect senses only provide indirect access to the real world, i.e. to what Kant calls das Ding an sich (the thing-in-itself) . In addition, whenever the truth of a proposition is expressed, it is always done by an individual, and it can never be considered to represent a general and objective belief. These philosophical ideas are directly reflected in the mathematical formalism of subjective logic.

Subjective opinions Subjective opinions express subjective beliefs about the truth of state values/propositions with degrees of epistemic uncertainty, and can explicitly indicate the source of belief whenever required. An opinion is usually denoted as ω X A {\displaystyle \omega _{X}^{A}} where A {\displaystyle A\,\!} is the source of the opinion, and X {\displaystyle X\,\!} is the state variable to which the opinion applies. The variable X {\displaystyle X\,\!} can take values from a domain (also called state space) e.g. denoted as X {\displaystyle \mathbb {X} } . The values of a domain are assumed to be exhaustive and mutually disjoint, and sources are assumed to have a common semantic interpretation of a domain. The source and variable are attributes of an opinion. Indication of the source can be omitted whenever irrelevant.

Binomial opinions Let x {\displaystyle x\,\!} be a state value in a binary domain. A binomial opinion about the truth of state value x {\displaystyle x\,\!} is the ordered quadruple ω x = ( b x , d x , u x , a x ) {\displaystyle \omega _{x}=(b_{x},d_{x},u_{x},a_{x})\,\!} where:

These components satisfy b x + d x + u x = 1 {\displaystyle b_{x}+d_{x}+u_{x}=1\,\!} and b x , d x , u x , a x ∈ [ 0 , 1 ] {\displaystyle b_{x},d_{x},u_{x},a_{x}\in [0,1]\,\!} . The characteristics of various opinion classes are listed below.

The projected probability of a binomial opinion is defined as P x = b x + a x u x {\displaystyle \mathrm {P} _{x}=b_{x}+a_{x}u_{x}\,\!} .

Binomial opinions can be represented on an equilateral triangle as shown on figure. A point inside the triangle represents a ( b x , d x , u x ) {\displaystyle (b_{x},d_{x},u_{x})\,\!} triple. The b,d,u-axes run from one edge to the opposite vertex indicated by the Belief, Disbelief or Uncertainty label. For example, a strong positive opinion is represented by a point towards the bottom right Belief vertex. The base rate, also called the prior probability, is shown as a red pointer along the base line, and the projected probability, P x {\displaystyle \mathrm {P} _{x}\,\!} , is formed by projecting the opinion onto the base, parallel to the base rate projector line. Opinions about three values/propositions X, Y and Z are visualized on the triangle to the left, and their equivalent Beta PDFs (Probability Density Functions) are visualized on the plots to the right. The numerical values and verbal qualitative descriptions of each opinion are also shown.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Subjective logic

Start with the simplest possible case. Write down what Subjective logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Subjective logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Subjective logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Subjective logic

In research: Subjective logic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Subjective logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Subjective logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Bayesian statistics, Non-classical logic, so understanding it makes those chapters shorter.
In everyday life: Look for Subjective logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Subjective logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Subjective logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Subjective logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Subjective logic in simple terms?

Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and relatively unreliable sources.

Why does Subjective logic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Subjective logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Subjective logic.