Inverse-Wishart distribution

Key takeaways

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

Reference excerpt

In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution. We say X {\displaystyle \mathbf {X} } follows an inverse Wishart distribution, denoted as X ∼ W − 1 ( Ψ , ν ) {\displaystyle \mathbf {X} \sim {\mathcal {W}}^{-1}(\mathbf {\Psi } ,\nu )} , if its inverse X − 1 {\displaystyle \mathbf {X} ^{-1}} has a Wishart distribution W ( Ψ − 1 , ν ) {\displaystyle {\mathcal {W}}(\mathbf {\Psi } ^{-1},\nu )} . Important identities have been derived for the inverse-Wishart distribution.

Density The probability density function of the inverse Wishart is:

f X ( X ; Ψ , ν ) = | Ψ | ν / 2 2 ν p / 2 Γ p ( ν 2 ) | X | − ( ν + p + 1 ) / 2 e − 1 2 tr ⁡ ( Ψ X − 1 ) {\displaystyle f_{\mathbf {X} }({\mathbf {X} };{\mathbf {\Psi } },\nu )={\frac {\left|{\mathbf {\Psi } }\right|^{\nu /2}}{2^{\nu p/2}\Gamma _{p}({\frac {\nu }{2}})}}\left|\mathbf {X} \right|^{-(\nu +p+1)/2}e^{-{\frac {1}{2}}\operatorname {tr} (\mathbf {\Psi } \mathbf {X} ^{-1})}}

where X {\displaystyle \mathbf {X} } and Ψ {\displaystyle {\mathbf {\Psi } }} are p × p {\displaystyle p\times p} positive definite matrices, | ⋅ | {\displaystyle |\cdot |} is the determinant, and Γ p ( ⋅ ) {\displaystyle \Gamma _{p}(\cdot )} is the multivariate gamma function.

Theorems

Distribution of the inverse of a Wishart-distributed matrix If X ∼ W ( Σ , ν ) {\displaystyle {\mathbf {X} }\sim {\mathcal {W}}({\mathbf {\Sigma } },\nu )} and Σ {\displaystyle {\mathbf {\Sigma } }} is of size p × p {\displaystyle p\times p} , then A = X − 1 {\displaystyle \mathbf {A} ={\mathbf {X} }^{-1}} has an inverse Wishart distribution A ∼ W − 1 ( Σ − 1 , ν ) {\displaystyle \mathbf {A} \sim {\mathcal {W}}^{-1}({\mathbf {\Sigma } }^{-1},\nu )} .

Marginal and conditional distributions from an inverse Wishart-distributed matrix Suppose A ∼ W − 1 ( Ψ , ν ) {\displaystyle {\mathbf {A} }\sim {\mathcal {W}}^{-1}({\mathbf {\Psi } },\nu )} has an inverse Wishart distribution. Partition the matrices A {\displaystyle {\mathbf {A} }} and Ψ {\displaystyle {\mathbf {\Psi } }} conformably with each other

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Inverse-Wishart distribution

Start with the simplest possible case. Write down what Inverse-Wishart distribution claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Inverse-Wishart distribution 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 Inverse-Wishart distribution 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 Inverse-Wishart distribution

In research: Inverse-Wishart distribution appears in science 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 Inverse-Wishart distribution 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: Inverse-Wishart distribution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjugate prior distributions, Continuous distributions, Exponential family distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Inverse-Wishart distribution 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 Inverse-Wishart distribution in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Inverse-Wishart distribution 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 Inverse-Wishart distribution out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Inverse-Wishart distribution in simple terms?

In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

Why does Inverse-Wishart distribution matter?

Because it connects several science 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 Inverse-Wishart distribution?

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 Inverse-Wishart distribution.