Reference / Notation

Notation

Each symbol has one meaning across the documentation and the docstrings. The tables give the symbol, its meaning, the name the code uses, and the page that defines it. Indices are \(i\) for objects, \(k\) for bins, \(j\) for measured bins and \(s\) for draws; other indices are declared where they are used. The Glossary defines the terms.

Objects, bins and likelihoods

Symbol	Meaning	Code name	Where defined
\(N\)	number of objects	lk.N, model.N	model
\(d_i\)	data of object \(i\)	the inputs of Likelihoods.from_*	model
\(D\)	the data of all objects		model
\(z\)	true value of an object	values in result.sample_values()	model
\(K\)	number of bins	bins.K	model
\(B_k\)	bin \(k\)	bins.edges, bins.labels	model
\(v_k\)	volume of bin \(k\)	bins.volumes	model
\(u=T(z)\)	measure coordinate, in which the population is flat within a bin	Measure, bins.u_edges	inputs
\(\ell_i(z)\)	likelihood function of object \(i\), \(p(d_i\mid z)\)	density= of from_density	model
\(L_{ik}\)	bin-averaged likelihood	Likelihoods	model
\(\bar L_{ik}\)	\(L_{ik}\) normalised to a row maximum of one	lk.values	inputs
\(w_i\)	weight (multiplicity) of object \(i\)	lk.weights	model
\(W\)	total weight, \(\sum_iw_i\)	lk.W	model
\(\pi(z)\)	interim prior	interim=	model
\(P_{ik}\)	mass of object \(i\)’s posterior in bin \(k\)	the table of from_table(..., interim=)	model
\(\Pi_k\)	mass of the interim prior in bin \(k\)	Density.masses	model
\(\omega_s\)	importance weight of sample or draw \(s\)	sample_weights=	inputs
\(n_{\rm draw}\)	number of draws or importance weights	n_draws, n_proposals	engines

The population

Symbol	Meaning	Code name	Where defined
\(\rho_k\)	population fraction of bin \(k\)	result.samples, result.mean, est.rho	model
\(\Delta^{K-1}\)	the simplex: \(K\) non-negative numbers that sum to one		model
\(p(z\mid\rho)\)	population density: flat in the measure coordinate within each bin; its average over bin \(k\) is \(\rho_k/v_k\)	result.density() (the bin averages)	model
\(\lambda_k\)	rate: expected number of objects in bin \(k\)	result.rates, GammaRates	model
\(\Lambda\)	total expected number of objects	result.total	model
\(k_i\)	bin (label) of object \(i\)	result.sample_assignments()	model
\(r_{ik}(\rho)\)	responsibility: object \(i\)’s posterior over bins given \(\rho\)	model.responsibilities	model
\(P(k_i=k\mid D)\)	hierarchical posterior of object \(i\)	result.assignment_probabilities()	model
\(n_k\)	realised histogram: observed objects in bin \(k\)	result.realized_counts()	model
\(\hat n_k(\rho)\)	expected counts, \(\sum_iw_ir_{ik}(\rho)\)	model.expected_counts	model
\(g_k\)	KKT ratio \(W^{-1}\sum_iw_iL_{ik}/(L_i\cdot\hat\rho)\); with a flat prior and no selection the KKT gap is \(\max_kg_k-1\)	kkt_gap	point estimates
\(f\)	fraction of missing information	model.missing_information	point estimates
\(I_{\rm obs}\), \(I_c\)	observed and complete-data information		point estimates

Selection and responses

Symbol	Meaning	Code name	Where defined
\(S_k\)	selection: probability that an object in bin \(k\) is observed, or its exposure	Selection.values	model
\(\tilde L_{ik}\)	detected-object likelihood: \(S_kL_{ik}\) on the truth, \(L_{ik}\) on the measurement		model
\(S^{\rm truth}(z)\), \(S^{\rm meas}(d)\)	the two parts of a combined selection	Selection.combined(on_truth=, on_measurement=)	selection
\(\phi_k\)	selected population: \(S_k\rho_k/S\cdot\rho\)	result.selected	point estimates
\(\chi_k\)	composition of the rates in mix-SQP		point estimates
\(M\)	number of objects the selection missed		model
\(m_k\)	missed objects in bin \(k\)	realized_counts(full=True)	model
\(n^{\rm inj}_k\), \(n^{\rm rec}_k\)	injected and recovered objects in bin \(k\)	selection.injections	selection
\(\tau_S\)	step scale of the random walk on logit \(S\)	smooth= of from_injections	selection
\(J\)	number of measured bins	response.measured_bins.K	inputs
\(R_{jk}\)	response: probability of measured bin \(j\) and observation, given true bin \(k\)	response.matrix	inputs
\(\epsilon_k\)	efficiency: probability that an object in true bin \(k\) is observed	response.efficiency	inputs

Priors

Symbol	Meaning	Code name	Where defined
\(\alpha_k\)	Dirichlet concentration, also of a tallied response’s columns	Dirichlet(alpha), prior= of Response.from_counts	priors
\(n^{\rm ref}_k\)	counts of a reference sample	counts of Dirichlet.from_counts	priors
\(\kappa\)	weight of the reference counts in \(\alpha_k\)	strength=	priors
\(G_0\)	base measure of a prior	base=	priors
\(\nu_k\)	stick-breaking proportions of a generalised Dirichlet		priors
\(c_{\rm PT}\), \(p_{\rm PT}\), \(h\)	Pólya-tree concentration, its growth with depth, and the depth	PolyaTree(c, power)	priors
\(a_k\), \(b_k\)	Gamma shape and rate of the rate \(\lambda_k\)	GammaRates(shape, rate)	model
\(\eta\)	log-fractions, \(\rho=\mathrm{softmax}(\eta)\)		priors
\(\mu\)	prior mean of \(\eta\)	mean=	priors
\(E\)	basis of a logistic prior, one column per mode		priors
\(\theta\)	hyperparameters: amplitude, scale, length scale	amplitude=, scale=, length_scale=	priors
\(s_j(\theta)\)	prior standard deviation of mode \(j\)		priors
\(\xi_j\)	whitened coefficient of mode \(j\)		priors
\(c_j\)	coefficient of mode \(j\), \(s_j\xi_j\)		priors
\(\zeta_j\)	partially centred coefficient of mode \(j\)		engines
\(\gamma_j\)	partial-centring weight of mode \(j\)	info["centring"]	engines
\(\tau\)	standard deviation of the random walk’s differences	scale= of LogisticRandomWalk	priors
\(r\)	order of the random walk’s differences	order=	priors
\(e_j\)	eigenvalue \(j\) of the random walk’s difference precision		priors
\(\Delta\)	half the range of a grid axis, in the measure coordinate		priors
\(c_{\rm box}\)	size of the Gaussian-process box relative to \(\Delta\)	boundary_factor=	priors
\(n_{\rm basis}\)	number of sines per axis	n_basis=	priors
\(f_{\rm ker}\)	resolution factor of a kernel		defaults
\(\psi_k(z)\)	component \(k\) of a Basis	Basis	inputs
\(\beta_k\)	fraction of the population in component \(k\)	result.component_draws	inputs
\(A_{jk}\)	mass of component \(k\) in bin or cell \(j\)	basis.matrix	inputs

Coordinates, engines and diagnostics

Symbol	Meaning	Code name	Where defined
\(x\)	unconstrained coordinates	result.x	priors
\(Q\)	orthonormal Helmert basis of the isometric log-ratio		priors
\(H\)	Hessian of the log posterior		engines
\(\hat R\)	split R-hat	diagnostics.rhat, result.diagnostics["rhat"]	validation
ESS	effective sample size, bulk or tail	diagnostics.ess, result.diagnostics["ess_bulk"]	validation
\(\hat k\)	Pareto shape of importance weights	diagnostics.psis, result.info["k_hat"]	engines
\(\tau_{\rm int}\)	integrated autocorrelation time of a chain	diagnostics.autocorr_time	engines
\(V_i(\rho)\)	relative Monte Carlo variance of a likelihood from samples	lk.sample_variance	inputs
\(n_{\rm sim}\)	number of simulated data sets	n_simulations=	validation

Symbols on older pages

Some guides, tutorials and docstrings were written before this table and still use a symbol with a second meaning. Read them with this mapping; the section rewrites of the documentation adopt the symbols above and remove it.

Older symbol	Meaning there	Symbol here
\(S\)	number of importance proposals; number of simulations in SBC	\(n_{\rm draw}\); \(n_{\rm sim}\)
\(S\)	half the range of a grid axis	\(\Delta\)
\(\lambda\)	strength of a reference sample	\(\kappa\)
\(\lambda_j\)	partial-centring weight	\(\gamma_j\)
\(\ell\)	log-likelihood of the catalogue	\(\log p(D\mid\rho)\)
\(\ell_{ik}\)	likelihood normalised over the bins	\(L_{ik}/\sum_kL_{ik}\)
\(w_k\)	weight of a Basis component	\(\beta_k\)
\(w_s\)	importance weight of a sample	\(\omega_s\)
\(\phi_k(z)\)	component of a Basis	\(\psi_k(z)\)
\(f(z)\)	population density	\(p(z\mid\rho)\)
\(\psi\)	composition of the rates in mix-SQP	\(\chi\)
\(B\)	basis of a logistic prior	\(E\)
\(c\), \(m\), \(p\)	Pólya-tree concentration, depth and growth	\(c_{\rm PT}\), \(h\), \(p_{\rm PT}\)
\(f\)	resolution factor of a kernel	\(f_{\rm ker}\)
\(m\)	reference counts	\(n^{\rm ref}_k\)
\(M\)	draws per SBC fit; component masses on a grid	\(n_{\rm draw}\); \(A\)
\(n_k\), \(d_k\)	injected and recovered objects	\(n^{\rm inj}_k\), \(n^{\rm rec}_k\)
\(z_i\)	bin of object \(i\)	\(k_i\)
\(x_i\)	data of object \(i\)	\(d_i\)
\(s_i\)	\(L_i\cdot\rho\)	\(L_i\cdot\rho\)
\(V\)	Helmert basis	\(Q\)
\(V_k\)	stick-breaking proportion	\(\nu_k\)
\(C(z)\), \(T(d)\)	parts of a combined selection	\(S^{\rm truth}(z)\), \(S^{\rm meas}(d)\)
\(\tau\)	step scale of a smooth injection selection	\(\tau_S\)
\(\beta\)	concentration of a tallied response	\(\alpha\)
\(G\)	gradient of the log posterior in \(\lambda\)	\(\partial\log p/\partial\lambda\)

The field pages quote each field’s own formulas. On the gravitational-wave page, \(\Lambda\) denotes hyperparameters, \(\theta\) an event’s parameters, \(S_i\) the number of samples of event \(i\), \(R\) the merger rate and \(N_\text{draw}\) the number of injections. On the empirical-Bayes page, \(w\) and \(g\) denote \(\rho\).