Priors and Parameter Initialization

Every free parameter carries a prior — a probability distribution over its values. The prior does double duty: a Bayesian sampler uses it as the prior in the usual sense, and an optimizer uses it as the distribution its initial population is drawn from. A prior is an orthogonal distribution family × scale: the family (Normal, Uniform, Gamma, …) fixes the shape, the scale (linear or base-10 logarithmic) fixes the space the parameter is sampled, proposed, and stored in. The family is always evaluated in that scale.

The *_var keyword

A prior is declared with a *_var keyword whose name encodes the family and the scale. The naming is regular: a family with base b yields b_var (linear) and logb_var (log10). So the Normal family gives normal_var and lognormal_var; Gamma gives gamma_var and loggamma_var; and so on. The value gives the parameter id followed by the family’s parameters:

uniform_var  = k1__FREE 0.01 100        # id, lower, upper
normal_var   = k2__FREE 1.0 0.3         # id, mean, sd
gamma_var    = k3__FREE 2.0 0.5         # id, shape, scale
exponential_var = k4__FREE 1.0          # id, scale (one-parameter family)

The full per-keyword syntax is under the configuration reference.

Parameter scale

The scale is the space the parameter is sampled, proposed, and stored in — linear or log10. The prior and the proposal arithmetic share it, and the posterior target is defined directly in this scale with no change of variables. The log prefix on a keyword selects log10; a bare “log” always means log10 across PyBNF, matching the noise-model additive scale. Log-scale priors are the right choice for a rate constant or concentration that ranges over orders of magnitude. (The natural-log scale is reachable only through the labelled parameter record below, not the positional *_var grammar.)

Support and reflecting bounds

A family’s support — the region where its density is nonzero — is intrinsic to the family: Uniform is finite, Normal and Laplace are unbounded, the positive families (Gamma, Exponential, half-Normal, …) are bounded below at zero, and Beta is bounded at both ends of [0, 1].

Reflecting bounds are a separate idea. They are a box a proposal is folded back into during a fit. A parameter gets one from any of three places:

  • The family’s own support. Outside it the declared density is exactly zero, so there is nothing there for a fit to find, and the support is always a wall — a gamma_var parameter is reflected at 0 and a beta_var one at 0 and 1 whether or not you write any bounds. (Without this, a population optimizer, which does not add the prior to its objective, could spend simulations on — and report a best fit at — a negative rate constant.)

  • Declared truncation. lower: / upper: on a parameter record truncate the prior to a box, renormalizing its density over that box and reflecting proposals off it. A declared bound may only tighten the support’s own wall: one outside the support is refused, since it would put a wall in the zero-density region (almost always a wrong family or scale).

  • The Uniform box. For uniform_var / loguniform_var the support is the box, and it alone takes an optional trailing flag — b (or blank) keeps the parameter bounded, while u turns the reflection off, letting the search leave the range it was seeded from:

    uniform_var = x__FREE 10 30 u    # sample in [10, 30], but allow moves outside

    No other family takes the flag: the wall of a support or a declared truncation is where the density ends, not a seeding range, so there is nothing to switch off.

Two samplers keep a bounded parameter inside the box by another route: adaptive MCMC (Adaptive MCMC) and DREAM (DREAM) reject a proposal that would leave it, and the chain stays where it is for that iteration. Both propose along directions that correlate the parameters, and folding such a proposal back into the box, one parameter at a time, would distort the distribution they sample.

A support wall is not a box for the purposes of the start-point optimizers (Simplex, Powell, CMA-ES, the gradient methods): those search a finite box, and the half-line a gamma_var declares is not one, so they refuse it. Give such a parameter a lower: / upper: box (or a uniform_var prior) to search it with those methods.

Half-bounded boxes: one side open

Writing one side and leaving the other open — lower: 1e-12, upper: inf — is a truncation, not a support wall, and the start-point optimizers take it. The search is confined to the half-line, which is what the reflecting box is; the start is the truncated prior’s median, which is finite; and a start_point line pins it wherever you like.

The one thing a half-line cannot supply is a width, and CMA-ES needs one per coordinate to scale its first population (its initial per-coordinate standard deviation is cmaes_sigma0 times the width). For an open side the width is taken from the central 80% of the coordinate’s own truncated prior — ppf(0.9) - ppf(0.1), measured in the parameter’s sampling space — so it is a real length in the parameter’s units, derived from the distribution you declared. It is the prior’s spread, though, not a range you stated: if you have a range in mind, writing both sides is more explicit and is what the optimizer will use.

Distribution families

The families below are all reachable through the positional *_var grammar. Each row lists the linear keyword; every family also has the log-prefixed log10 form (lognormal_var, loggamma_var, …).

Family (linear keyword)

Parameters

Support

Notes

uniform_var

lower, upper

finite box

The box-bounded prior; takes the b / u reflecting-bounds flag. loguniform_var is the log10 form.

normal_var

mean, sd

\((-\infty, \infty)\)

Gaussian.

laplace_var

location, scale

\((-\infty, \infty)\)

Heavier-tailed than Normal.

cauchy_var

location, scale

\((-\infty, \infty)\)

Very heavy tails.

gumbel_var

location, scale

\((-\infty, \infty)\)

Extreme-value.

logistic_var

location, scale

\((-\infty, \infty)\)

Symmetric, slightly heavier-tailed than Normal.

gamma_var

shape, scale

\((0, \infty)\)

Positive; PEtab-catalog parity.

inv_gamma_var

shape, scale

\((0, \infty)\)

Conjugate prior for a variance.

weibull_var

shape, scale

\((0, \infty)\)

Lifetime / time-to-event.

beta_var

alpha, beta

\([0, 1]\)

The canonical prior for a fraction.

exponential_var

scale

\((0, \infty)\)

One-parameter.

chisquare_var

dof

\((0, \infty)\)

One-parameter.

rayleigh_var

scale

\((0, \infty)\)

One-parameter.

half_normal_var

scale

\((0, \infty)\)

The right half of a zero-centered Normal; a mild positive scale prior.

half_cauchy_var

scale

\((0, \infty)\)

The right half of a zero-centered Cauchy; a weakly-informative scale prior.

The positive-support and log-scale families are natural priors for an estimated noise parameter — a standard deviation or dispersion that must stay positive.

No prior: start points

The keywords var (linear) and logvar (log10) give a parameter a single start value and no prior distribution. They are the start points for the start-point optimizers — Simplex, Powell, and CMA-ES. A no-prior parameter still carries a scale and is varied during the fit; it simply contributes nothing to the log prior and cannot be prior-sampled:

var    = k__FREE 1.5
logvar = k__FREE 0.001

Multi-parameter priors: the parameter record

The positional *_var grammar carries at most two distribution parameters, so a family with three — like Student-t (degrees of freedom, location, scale) — has no positional keyword. These are authored instead through the edition-2 labelled parameter: record, which names each field, and which is also where the natural-log parameter scale is selected. Tutorial lesson 32_prior_gallery walks through the full catalog, including the Student-t record.

See also