mathematics

Laplace's approximation

Laplace's approximation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Laplace's approximation rather than just read about it. In short: Laplace's approximation or the quadratic approximation (QUAP) provides an analytical expression for a posterior probability distribution by fitting a Gaussian distribution with a mean equal to the MAP solution and precision equal to the observed Fisher information. The approximation is justified by the Bernstein–von Mises theorem, which states that, under regularity conditions, the error of the approximation tends t…

Key takeaways

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

Reference excerpt

Laplace's approximation or the quadratic approximation (QUAP) provides an analytical expression for a posterior probability distribution by fitting a Gaussian distribution with a mean equal to the MAP solution and precision equal to the observed Fisher information. The approximation is justified by the Bernstein–von Mises theorem, which states that, under regularity conditions, the error of the approximation tends to 0 as the number of data points tends to infinity. For example, consider a regression or classification model with data set { x n , y n } n = 1 , … , N {\displaystyle \{x_{n},y_{n}\}_{n=1,\ldots ,N}} comprising inputs x {\displaystyle x} and outputs y {\displaystyle y} with (unknown) parameter vector θ {\displaystyle \theta } of length D {\displaystyle D} . The likelihood is denoted p ( y | x , θ ) {\displaystyle p({\bf {y}}|{\bf {x}},\theta )} and the parameter prior p ( θ ) {\displaystyle p(\theta )} . Suppose one wants to approximate the joint density of outputs and parameters p ( y , θ | x ) {\displaystyle p({\bf {y}},\theta |{\bf {x}})} . Bayes' formula reads:

p ( y , θ | x ) = p ( y | x , θ ) p ( θ | x ) = p ( y | x ) p ( θ | y , x ) ≃ q ~ ( θ ) = Z q ( θ ) . {\displaystyle p({\bf {y}},\theta |{\bf {x}})\;=\;p({\bf {y}}|{\bf {x}},\theta )p(\theta |{\bf {x}})\;=\;p({\bf {y}}|{\bf {x}})p(\theta |{\bf {y}},{\bf {x}})\;\simeq \;{\tilde {q}}(\theta )\;=\;Zq(\theta ).}

The joint is equal to the product of the likelihood and the prior and by Bayes' rule, equal to the product of the marginal likelihood p ( y | x ) {\displaystyle p({\bf {y}}|{\bf {x}})} and posterior p ( θ | y , x ) {\displaystyle p(\theta |{\bf {y}},{\bf {x}})} . Seen as a function of θ {\displaystyle \theta } the joint is an un-normalised density. In Laplace's approximation, we approximate the joint by an un-normalised Gaussian q ~ ( θ ) = Z q ( θ ) {\displaystyle {\tilde {q}}(\theta )=Zq(\theta )} , where we use q {\displaystyle q} to denote approximate density, q ~ {\displaystyle {\tilde {q}}} for un-normalised density and Z {\displaystyle Z} the normalisation constant of q ~ {\displaystyle {\tilde {q}}} (independent of θ {\displaystyle \theta } ). Since the marginal likelihood p ( y | x ) {\displaystyle p({\bf {y}}|{\bf {x}})} doesn't depend on the parameter θ {\displaystyle \theta } and the posterior p ( θ | y , x ) {\displaystyle p(\theta |{\bf {y}},{\bf {x}})} normalises over θ {\displaystyle \theta } we can immediately identify them with Z {\displaystyle Z} and q ( θ ) {\displaystyle q(\theta )} of our approximation, respectively. Laplace's approximation is

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Laplace's approximation

Start with the simplest possible case. Write down what Laplace's approximation 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 Laplace's approximation 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 Laplace's approximation 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 Laplace's approximation

In research: Laplace's approximation 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 Laplace's approximation 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: Laplace's approximation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Bayesian inference, Statistical approximations, so understanding it makes those chapters shorter.
In everyday life: Look for Laplace's approximation 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.

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How to study Laplace's approximation in 20 minutes

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

Frequently asked questions

What is Laplace's approximation in simple terms?

Why does Laplace's approximation 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 Laplace's approximation?

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 Laplace's approximation.