Interactive explorer

Four methods, one posterior

The four algorithms from the assignment, ported line by line from R and run in your browser. With the original settings they reproduce R’s output exactly (and, in As submitted mode, the numbers printed in the 2023 report); change the seed or the tuning to see how each road behaves.

The same four R functions, fed the data the assignment describes: 7 doses, 10 patients each, and logit p = β₀ + β₁·dose. All four roads should arrive at nearly the same posterior.

Model and data

What every method is fitting

\begin{aligned} y_i \mid \beta &\sim \operatorname{Bin}(n_i,\, p_i), \qquad n_i = 10 \\ \log\frac{p_i}{1-p_i} &= \beta_0 + \beta_1\, x_i, \qquad x_i \in \{0,\dots,6\} \\ p(\beta_0, \beta_1) &\propto 1 \end{aligned}

Seven grouped binomial observations, one per dose level, with a flat prior so the posterior is proportional to the likelihood. Two coefficients: an intercept and the change in log-odds of improvement per unit of dose.

Patients improved at each dose
dose $x_i$	0	1	2	3	4	5	6
improved $y_i$	3	5	4	4	7	6	8
not improved	7	5	6	6	3	4	2
proportion	0.3	0.5	0.4	0.4	0.7	0.6	0.8

Each method runs in a Web Worker with R’s Mersenne-Twister seeded at 90125.

Laplace approximation
approximation
Waiting…
Metropolis-Hastings
sampler
Waiting…
Hamiltonian Monte Carlo
sampler
Waiting…
Expectation propagation
approximation
Waiting…

Marginal posteriors

Four estimates of β₁

Kernel density of the sampler draws (solid) against the two Gaussian approximations (dashed and dotted). Hover to read the curves.

Laplace approximation
Metropolis-Hastings
Hamiltonian Monte Carlo
Expectation propagation

Scale each curve to its own peak

Fitting…

Posterior summaries of β₁ dose for each method
Method	mean	sd	95% interval	ESS
Laplace	…	…	…	…
MH	…	…	…	…
HMC	…	…	…	…
EP	…	…	…	…

Intervals are 2.5% and 97.5% quantiles of the draws for MH and HMC, and mean ± 1.96 sd for the Gaussian approximations. ESS is coda::effectiveSize, reported as is: it can exceed the number of draws when successive HMC draws are negatively correlated.

Mixing

Trace plots for β₁

Every draw from the start of each chain (thinned for drawing). The shaded strip is the burn-in the original code discarded.

Metropolis-Hastings

Hamiltonian Monte Carlo

Joint posterior

β₀ against β₁

Grey contours: the exact posterior (50/80/95/99% Gaussian-equivalent levels). Dots: MH and HMC draws. Ellipses: 50% and 95% regions of Laplace and EP.

Laplace
MH
HMC
EP

What it means

Probability of improvement by dose

Posterior median and 95% band of P(improvement) implied by each method. Open circles are the observed proportions out of 10.

Laplace
MH
HMC
EP

Every number, every coefficient

Posterior summaries for every coefficient and method
Coefficient	Method	mean	sd	2.5%	50%	97.5%	ESS
β₀ (Intercept)	Laplace	…	…	…	…	…	…
	MH	…	…	…	…	…	…
	HMC	…	…	…	…	…	…
	EP	…	…	…	…	…	…
β₁ dose	Laplace	…	…	…	…	…	…
	MH	…	…	…	…	…	…
	HMC	…	…	…	…	…	…
	EP	…	…	…	…	…	…