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Inclusion of endpoints does not change its posterior probability. Every point inside has higher posterior density than every admissible point to its right, and points below

c

have density zero. This is the Pareto highest density interval; equal endpoint densities are unnecessary because one endpoint is the support boundary.

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Past exam of the statistics course of the University of Oxford / 2021 / 1 / a / iii / Solution 2026-10-25

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Write

A = - \sum_{i} lo g X_{i} > 0

, with

lo g

the natural logarithm. The likelihood function and log-likelihood function are

L (θ) = θ^{n} exp {- (θ - 1) A}, ℓ (θ) = n lo g θ - (θ - 1) A .

(1)

The score function is

ℓ^{'} (θ) = n / θ - A

and the second derivative is

ℓ^{''} (θ) = - n / θ^{2} < 0

. Thus the log-likelihood function is strictly concave, its stationary point is the unique global maximum, and the maximum-likelihood estimator is

T_{2} = θ = \frac{n}{A} = - \frac{n}{\sum _{i} lo g X _{i}} .

(2)

The one-observation Fisher information is

I (θ) = 1/ θ^{2}

. The support of a probability distribution here is independent of

θ > 0

, so the regular asymptotic normality of a maximum likelihood estimator gives

n (T_{2} - θ) d N (0, θ^{2}), T_{2} \tilde{˙} N (θ, θ^{2} / n), v (T_{2}) = θ^{2} / n .

(3)

There is also a direct check:

U_{i} = - lo g X_{i}

has an exponential distribution with rate

θ

, since

P (U_{i} > u) = P (X_{i} < e^{- u}) = e^{- θ u}

for

u > 0

. Its mean and variance are

1/ θ

and

1/ θ^{2}

. Apply the central limit theorem to

A / n

and the delta method to

h (u) = 1/ u

to obtain the same result.

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Past exam of the statistics course of the University of Oxford / 2021 / 1 / b / i / Solution 2026-10-25

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Put

A = - \sum_{i} lo g x_{i} > 0

B = - \sum_{j} lo g y_{j} > 0

, and

N = n + m

. The two independent statistical samples give the joint likelihood function

L (θ, τ) = θ^{n} τ^{m} exp {- (θ - 1) A - (τ - 1) B} .

(1)

The unrestricted maximum-likelihood estimators are

θ = n / A

and

τ = m / B

. Under the null hypothesis, the common statistical parameter

η

has log-likelihood function

N lo g η - (η - 1) (A + B)

, maximized at

η = N / (A + B)

. With the likelihood ratio defined as the null maximum divided by the unrestricted maximum,

λ = \frac{( N / ( A + B ) ) ^{N} e ^{- N + A + B}}{( n / A ) ^{n} ( m / B ) ^{m} e ^{- N + A + B}} = \frac{N ^{N}}{n ^{n} m ^{m}} \frac{A ^{n} B ^{m}}{( A + B ) ^{N}} .

(2)

The statistic in the paper is

T = A / (A + B)

, since both original sums of natural logarithms are negative. Hence

λ = c (n, m) T^{n} (1 - T)^{m}, c (n, m) = \frac{( n + m ) ^{n + m}}{n ^{n} m ^{m}} .

(3)

It has maximum one at

T = n / N

, corresponding to equal fitted rates. This is the likelihood-ratio test for equality of beta shape parameters.

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Past exam of the statistics course of the University of Oxford / 2021 / 3 / a / i / Solution 2026-10-25

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Set

S = \sum_{i} x_{i}

and

N = nm

. Conditional independence of the binomial distribution observations gives the likelihood function

L (θ) = {\prod_{i = 1}^{n} (x _{i} m)} θ^{S} (1 - θ)^{N - S} .

(1)

The prior distribution is the beta distribution with parameters

(1, β)

. By Bayes' theorem, multiplying the prior density by the likelihood function gives a posterior density proportional to

θ^{S} (1 - θ)^{N - S + β - 1} .

(2)

Recognizing and normalizing this beta distribution gives

θ ∣ x \sim Beta (S + 1, N - S + β), π (θ ∣ x) = \frac{θ ^{S} ( 1 - θ ) ^{N - S + β - 1}}{B ( S + 1 , N - S + β )}

(3)

for

0 < θ < 1

. This is Beta-binomial conjugacy; the original binomial coefficients cancel in the normalization because they are independent of

θ

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Past exam of the statistics course of the University of Oxford / 2021 / 3 / b / i / Solution 2026-10-25

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Let

S_{xx} = \sum_{i} x_{i}^{2}

S_{x y} = \sum_{i} x_{i} y_{i}

, and

\overset{y}{ˉ} = n^{- 1} \sum_{i} y_{i}

. The independent observations in the normal linear model give

L (α, β) \propto exp {- \frac{1}{2 σ ^{2}} \sum_{i} (y_{i} - α - β x_{i})^{2}} .

(1)

The centered design condition

\sum_{i} x_{i} = 0

removes the

α β

term:

\sum_{i} (y_{i} - α - β x_{i})^{2} = \sum_{i} y_{i}^{2} + n α^{2} + S_{xx} β^{2} - 2 n \overset{y}{ˉ} α - 2 S_{x y} β .

(2)

The stated prior density factors into independent normal distributions for the regression intercept

α

and regression coefficient

β

. Multiplication by the likelihood function therefore leaves separate quadratic expressions in the two coefficients. Define the positive posterior variances

v_{α} = (\frac{n}{σ ^{2}} + \frac{1}{τ _{0}^{2}})^{- 1}, v_{β} = (\frac{S _{xx}}{σ ^{2}} + \frac{1}{τ _{1}^{2}})^{- 1},

(3)

and the posterior means

m_{α} = v_{α} (\frac{n y ˉ}{σ ^{2}} + \frac{α _{0}}{τ _{0}^{2}}), m_{β} = v_{β} (\frac{S _{x y}}{σ ^{2}} + \frac{β _{0}}{τ _{1}^{2}}) .

(4)

By completing the square, the joint posterior density is

π (α, β ∣ y) = \frac{1}{2 π v _{α} v _{β}} exp {- \frac{( α - m _{α} ) ^{2}}{2 v _{α}} - \frac{( β - m _{β} ) ^{2}}{2 v _{β}}} .

(5)

Equivalently,

α ∣ y \sim N (m_{α}, v_{α}), β ∣ y \sim N (m_{β}, v_{β}), α ⊥ β ∣ y .

(6)

This is Gaussian conjugacy for a normal linear model. Each posterior precision parameter is the sum of its data and prior precision parameters. The formulas remain valid when

S_{xx} = 0

: all

x_{i}

are then zero, so the posterior distribution of

β

equals its prior distribution.

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Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 3 / 2 / b / ii / Solution 2026-10-20

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TODO: Complete the data-specific numerical results and diagnostics when the original Iris_data.csv is available.

The identity

\sum_{i} (x_{i} - μ)^{2} = \sum_{i} (x_{i} - \overset{x}{ˉ})^{2} + n (μ - \overset{x}{ˉ})^{2}

(1)

shows that the likelihood function is a bell-shaped curve in

μ

, maximized at the sample mean, with width

1/ n

. Its actual vertical scale also depends on the within-sample sum of squares.

Continue the R script with:

sse <- sum((x - xbar)^2)
loglik <- function(mu) {
  -n/2 * log(2*pi) - 0.5 * (sse + n * (mu - xbar)^2)
}
grid <- seq(3, 6, length.out = 2001L)
ell <- loglik(grid)
png("q2-likelihood.png", width = 900, height = 600)
par(mar = c(4.5, 4.5, 4, 1))
if (max(ell) > log(.Machine$double.xmin)) {
  plot(grid, exp(ell), type = "l", xlab = expression(mu),
       ylab = expression(L(mu)), main = "Likelihood on [3,6]")
} else {
  # Explicitly label a proportional likelihood if the true scale underflows.
  plot(grid, exp(ell - max(ell)), type = "l", xlab = expression(mu),
       ylab = "Likelihood relative to maximum on plotted grid",
       main = "Likelihood on [3,6] (rescaled)")
}
abline(v = xbar, col = "steelblue", lty = 2)
dev.off()

Working first with the log-likelihood avoids premature numerical underflow. The fallback preserves the shape on the requested interval and labels the altered scale. A data-specific plot requires the missing CSV file; no fabricated observation set is used here.

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Pedigree likelihood 2026-10-13

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For latent genotypes

G_{i}

and observed phenotypes

X_{i}

, assume conditional phenotype independence, independent founder genotypes, and conditionally independent Mendelian transmissions. The likelihood function is

\sum_{(G_{i})} \prod_{i} P (X_{i} ∣ G_{i}) \prod_{i \in F} P (G_{i}) \prod_{i \in / F} P (G_{i} ∣ G_{m (i)}, G_{f (i)}) .

(1)

This follows by the chain rule applied to the acyclic ancestry graph, then the law of total probability over all latent genotypes. Penetrance supplies the phenotype factors and genetic recombination supplies multilocus transmission factors.

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Phylogenetic pruning algorithm 2026-10-20

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A recursive calculation of the likelihood function on a phylogenetic tree, summing over unknown ancestral states while reusing partial calculations from descendant subtrees.^[ref]

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Piecewise-constant covariate approximation 2026-10-06

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Approximate changing covariates by constant values on observation intervals. The generator matrix is then constant on each interval, with transition matrix

exp (Q_{j} Δ t_{j})

. Multiplying the relevant entries gives a conditional likelihood function for discretely observed states.

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Poisson change-point posterior Created 2026-10-06 Updated 2026-10-14

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For an Inhomogeneous Poisson process whose known positive intensity changes from

λ_{1}

λ_{2}

at an unknown time

θ \in (0, T)

, the ordered arrival-time likelihood function is

L (θ) = λ_{1}^{j (θ)} λ_{2}^{n - j (θ)} e^{- λ_{1} θ - λ_{2} (T - θ)} .

(1)

Under a uniform prior, each interval between successive arrivals has an exponential posterior density with rate in the exponent

λ_{2} - λ_{1}

. Its integrated weights allow exact interval selection followed by a truncated exponential draw; when the two rates coincide, the Bayesian posterior is uniform and the data contain no change-time information.

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Poisson-gamma conjugacy Created 2026-09-24 Updated 2026-10-25

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For conditionally independent Poisson observations of mean

θ

and a gamma distribution prior of shape

r > 0

and rate

λ > 0

, the likelihood function contributes

θ^{\sum_{i} x_{i}} e^{- n θ}

. Multiplying by the prior density gives a posterior distribution with shape

r + \sum_{i} x_{i}

and rate

λ + n

. The single-observation update is shape

r + X

and rate

λ + 1

The posterior mean is the convex combination

\frac{r + \sum _{i} x _{i}}{λ + n} = \frac{λ}{λ + n} \frac{r}{λ} + \frac{n}{λ + n} \overset{x}{ˉ} .

(1)

Thus the prior mean

r / λ

and the sample maximum-likelihood estimate

\overset{x}{ˉ}

bracket it. Integrating the next conditional Poisson distribution against this posterior distribution gives the Poisson-gamma mixture, a negative binomial distribution; its spread includes uncertainty in

θ

as well as the conditional count variation.

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Pseudolikelihood 2026-10-20

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A pseudolikelihood multiplies conditional probabilities rather than the full joint likelihood function. It can be useful for fitting a Markov random field, but using the product itself as a probability density defines a different model unless an additional equivalence is established.

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Pseudo-marginal Metropolis–Hastings algorithm Created 2026-10-06 Updated 2026-10-14

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A pseudo-marginal Metropolis–Hastings algorithm replaces an intractable likelihood function by a nonnegative unbiased likelihood estimator and includes its auxiliary randomness in the state. If

E [L (θ, U)] = L (θ)

for

U \sim m_{θ}

, the extended target is proportional to

p (θ) L (θ, u) m_{θ} (u)

. An independence proposal for new auxiliary randomness cancels the

m_{θ}

factors in the Metropolis–Hastings acceptance probability. On rejection the old estimate must be retained. Integrating the extended target gives the desired marginal posterior distribution exactly.

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