Solomonoff's theory of inductive inference

Key takeaways

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

Reference excerpt

Solomonoff's theory of inductive inference in philosophy is a method of evaluating scientific models according to their description length. According to the theory, the best possible model is the shortest algorithm that generates the empirical data under consideration. In addition to the choice of data, other assumptions are that, to avoid the post-hoc fallacy, the programming language must be chosen prior to the data and that the environment being observed is generated by an unknown algorithm. This is also called a theory of induction. Due to its basis in the dynamical (state-space model) character of Algorithmic Information Theory, it encompasses statistical as well as dynamical information criteria for model selection. It was introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory. Solomonoff proved that this induction is incomputable (or more precisely, lower semi-computable), but noted that "this incomputability is of a very benign kind", and that it "in no way inhibits its use for practical prediction" (as it can be approximated from below more accurately with more computational resources). It is only "incomputable" in the benign sense that no scientific consensus is able to prove that the best current scientific theory is the best of all possible theories. However, Solomonoff's theory does provide an objective criterion for deciding among the current scientific theories explaining a given set of observations. Solomonoff's induction naturally formalizes Occam's razor by assigning larger prior credences to theories that require a shorter algorithmic description.

Origin

Philosophical The theory is based in philosophical foundations, and was founded by Ray Solomonoff around 1960. It is a mathematically formalized combination of Occam's razor and the Principle of Multiple Explanations. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.

Principle Solomonoff's induction has been argued to be the computational formalization of pure Bayesianism. To understand, recall that Bayesianism derives the posterior probability P [ T | D ] {\displaystyle \mathbb {P} [T|D]} of a theory T {\displaystyle T} given data D {\displaystyle D} by applying Bayes rule, which yields

P [ T | D ] = P [ D | T ] P [ T ] P [ D | T ] P [ T ] + ∑ A ≠ T P [ D | A ] P [ A ] {\displaystyle \mathbb {P} [T|D]={\frac {\mathbb {P} [D|T]\mathbb {P} [T]}{\mathbb {P} [D|T]\mathbb {P} [T]+\sum _{A\neq T}\mathbb {P} [D|A]\mathbb {P} [A]}}}

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Worked examples

Example 1 — a first encounter with Solomonoff's theory of inductive inference

Start with the simplest possible case. Write down what Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference

In research: Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference 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: Solomonoff's theory of inductive inference is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algorithmic information theory, Bayesian statistics, Epistemology, so understanding it makes those chapters shorter.
In everyday life: Look for Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Solomonoff's theory of inductive inference in simple terms?

Solomonoff's theory of inductive inference in philosophy is a method of evaluating scientific models according to their description length. According to the theory, the best possible model is the shortest algorithm that generates the empirical data under consideration.

Why does Solomonoff's theory of inductive inference 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 Solomonoff's theory of inductive inference?

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 Solomonoff's theory of inductive inference.