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Default priors

2026-10-26

Source: vignettes/default_priors.Rmd
default_priors.Rmd

This article documents the default priors in mcp. These are the priors used if you do not specify otherwise in mcp(..., prior = list()).

In short, mcp uses the conventional default priors of rstanarm for intercepts and coefficients and of brms for scale parameters, with mcp-specific defaults for change points, autocorrelation, and identity-link binomial models, and a few documented deviations.

See Working with priors for how to set your own priors and do prior predictive checks.

Default priors on regression parameters

mcp automatically assigns weakly informative default priors to the coefficients of all distributional parameters (mu, sigma, shape, and AR/MA). They rule out absurd values while staying wide enough that the data dominate in most applications. They are designed for stable estimation and prediction. You can inspect all resolved priors for any fitted model using prior_summary(fit, verbose = TRUE).

Priors for hypothesis tests should be informed by the problem at hand.

The default priors follow three principles:

  1. Coefficient priors are autoscaled as in rstanarm: For the mean (mu), \beta \sim \text{Normal}\left(0, \frac{2.5 \cdot s_y}{\operatorname{sd}(\text{column})}\right) where s_y = \operatorname{sd}(y) for identity links and s_y = 1 for other links (logit and probit links use 1.5 and 1 instead of 2.5; see deviations). \operatorname{sd}(\text{column}) is the SD of the coefficient’s model-matrix column, including the 0/1 dummies of factors. This makes the prior largely invariant to the units of x and y, e.g., whether time is measured in seconds or minutes. A priori, a change of one SD in a predictor changes the response (on the link scale) by less than 5 \cdot s_y with 95% probability.

    • For x-terms, \operatorname{sd}(x) is computed over all data, so slope priors do not depend on the number or location of change points. Powers such as I(x^2) use \operatorname{sd}((x - \min(x))^2). Interactions such as x:z multiply by the SD of the term without x, \operatorname{sd}(x) \cdot \operatorname{sd}(z), while 0/1 indicators such as the dummies in x:group contribute a factor of 1 (see deviations).

    • The autoscaling does not account for segment length. If you expect steep slopes or strong curvature within short segments, consider setting a wider prior.

  2. Intercepts follow rstanarm: \text{Normal}(\operatorname{mean}(y), 2.5 \cdot \operatorname{sd}(y)) for identity links, \text{Normal}(0, 1.5) for logit links, and \text{Normal}(0, 1) for probit links (see deviations for these and for log links). Similar to rstanarm’s prior on centered predictors, the default prior on Intercept_1 applies to the value at the start of the data, \min(x), rather than at x = 0. Intercept_1 itself is still the value at x = 0, as in lm(), so prior_summary() shows its prior given the slope, e.g., dnorm(mean(y) - x_1 * min(x), 2.5 * sd(y)). Later intercepts are already at their change point. See deviations for details.

  3. Scale parameters follow brms: The residual SD and, for identity links, group-level SDs get a half-Student-t, \text{Student-t}(3, 0, S_y) \, T(0, ) with S_y = \max(2.5, \operatorname{mad}(y)). Half-Student-t priors are a standard weakly informative choice for scale parameters, as recommended for group-level SDs by Gelman (2006).

Summary reference table

General quantities used in the table below:

  • \operatorname{sd}(\text{column}): the SD of the coefficient’s model-matrix column, as described above.

  • response scale S_y = \max(2.5, \operatorname{mad}(y))

  • log-rate scale S_{\text{rate}} = \max(2.5, \operatorname{mad}(y_{\text{rate}}))

  • log-rate y_{\text{rate}} = \log(y) - \text{offset}, with y \le 0 replaced by 0.1, as brms does for zeros (and \text{offset} = 0 if absent)

  • As in brms, \operatorname{mad}() and the \operatorname{median}() of log-rates are rounded to one decimal. prior_summary(fit) shows the resulting values.

Parameter & Link Intercept Coefficients Group SD (\sigma_{\text{group}}) Precedent
Mean (\mu): Identity link (gaussian, poisson*) dnorm(mean(y), 2.5 * sd(y)) dnorm(0, 2.5 * sd(y) / sd(column)) dt(0, S_y, 3) T(0, )† rstanarm (group SD: brms)
Mean (\mu): Log link (poisson, negbinomial, gaussian) dnorm(median(y_rate), S_rate) dnorm(0, 2.5 / sd(column)) dnorm(0, 2.5) T(0, )† rstanarm coefficients
Mean (\mu): Logit link (binomial, bernoulli) dnorm(0, 1.5) dnorm(0, 1.5 / sd(column)) dt(0, 1.5, 3) T(0, )† rstanarm with mcp-specific scales
Mean (\mu): Probit link (binomial, bernoulli) dnorm(0, 1) dnorm(0, 1 / sd(column)) dt(0, 1, 3) T(0, )† rstanarm with mcp-specific scales
Mean (\mu): Identity link (binomial, bernoulli) dunif(0, 1) contrasts: dunif(-1, 1)
slopes: dnorm(0, 1 / sd(column)) 		dt(0, 1, 3) T(0, )†
Constant dispersion (implicit sigma, shape) sigma_1: dt(0, S_y, 3) T(0, )
shape_1: dinvgamma(0.4, 0.3) 		— — brms
Modeled sigma() (log link) dt(0, 2.5, 3) contrasts: dt(0, 2.5, 3)
slopes: dt(0, 2.5 / sd(column), 3) 		dt(0, 2.5, 3) T(0, )† brms intercept
Modeled shape() (log link) dnorm(0, 2.5) contrasts: dnorm(0, 2.5)
slopes: dnorm(0, 2.5 / sd(column)) 		dnorm(0, 2.5) T(0, )†
Autocorrelation (ar(), ma()) dnorm(0, 0.5) T(-1, 1) contrasts: dnorm(0, 0.25)
slopes: dnorm(0, 0.25 / sd(column)) 		—
Change points (cp_i) dunif(min(x), max(x))** — dnorm(0, (max(x) - min(x)) / (2 * (n_cp() + 1))) T(0, )

* For poisson(), these are rstanarm’s Gaussian defaults (see deviations), and the intercept is truncated T(0, ), or T(-min(offset), ) with an offset(). † Group SDs for slopes are divided by \operatorname{sd}(\text{column}). A blank precedent means that the default is mcp-specific. ** Ordered (flat Dirichlet)

Default prior on change points

The default prior on change points is a flat Dirichlet prior on the segment lengths, also known as “uniform order statistics”. It treats any set of ordered change points within the observed range of x as equally likely. Equivalently, the change points are distributed like sorted draws from \text{Uniform}(\min(x), \max(x)). This is also how fit$prior and the JAGS code write it: dunif(min(x), max(x)) on each change point, restricted to ordered change points. See Working with priors for how to set other priors on change points.

With K segments, i.e., K - 1 change points \tau_1 < \dots < \tau_{K-1}, this prior has some nice properties:

  • Beta marginals: Each change point has a beta prior, scaled to the observed range of x:

    \frac{\tau_k - \min(x)}{\max(x) - \min(x)} \sim \text{Beta}(k, K - k)

    For 1 change point, this is \text{Beta}(1, 1), i.e., \text{Uniform}(\min(x), \max(x)). For 2 change points, \tau_1 and \tau_2 have scaled \text{Beta}(1, 2) and \text{Beta}(2, 1) priors.

  • Evenly spaced on average: The prior means are

    \mathbb{E}[\tau_k] = \min(x) + \frac{k}{K} (\max(x) - \min(x))

    For example, with 2 change points, the prior means are at 33% and 67% of the observed range of x. With 5 change points, they are at 16.7%, 33.3%, 50%, 66.7%, and 83.3%.

  • Symmetry and flat sum: The prior is symmetric along x, and the sum of the prior densities of all change points is uniform across the observed range of x.

  • More change points, more informative: For a given range, adding change points makes the prior on each change point more informative because they are “squeezed” together.

The plot below compares these theoretical beta marginals (black lines) with prior draws from mcp (red lines) for 2 and 5 change points:

The Dirichlet prior on segment lengths

The change points divide the observed range of x into K segments. Instead of a prior on each change point, one can place a prior on how the range is divided, i.e., on the relative segment lengths

s_k = \frac{\tau_k - \tau_{k-1}}{\max(x) - \min(x)}, \qquad k = 1, \dots, K,

with \tau_0 = \min(x) and \tau_K = \max(x), the ends of the observed range. They are positive and sum to 1. The Dirichlet distribution is the standard prior for such proportions. It also underlies monotonic effects in brms (Bürkner & Charpentier, 2020). The symmetric Dirichlet prior with concentration \alpha has density

p(s_1, \dots, s_K) \propto \prod_{k=1}^{K} s_k^{\alpha - 1}.

With \alpha = 1, the density is constant, so every division of the range into ordered segments is equally likely. This is mcp’s default prior. With \alpha > 1, divisions with short segments are less likely, favoring evenly spaced change points away from the ends of the observed range. The marginal priors generalize to

\frac{\tau_k - \min(x)}{\max(x) - \min(x)} \sim \text{Beta}(k\alpha, (K - k)\alpha).

The segment lengths are a linear function of the change points, so the same density applies to the ordered change points. For \alpha > 1, the prior is therefore the default prior multiplied by the factor \prod_k s_k^{\alpha - 1}. This factor is 1 when all segments are equally long and approaches 0 when a segment becomes very short. See Working with priors for how to set \alpha and From formulas to sampler code for how JAGS samples it, including why \alpha < 1 is not supported.

Default priors on group-level effects

Each group-level effect has a population-level SD parameter. Group-level deviations in the predictor part use a zero-mean normal distribution governed by that SD, as in ordinary multilevel regression. With a multi-coefficient || term, every independent coefficient has its own SD. For identity links, the SD receives the brms half-Student-t prior on the response scale. Log, logit, and probit links and SDs inside sigma() and shape() use other scales (see the table above). Slope SDs are divided by \operatorname{sd}(\text{column}), like the population-level slopes. For distributional parameters, deviations and SD are on that parameter’s link scale (for example, effects inside sigma() are on the log-SD scale).

Group-level change points

Group-level change points use the same ordinary normal hierarchy, additionally bounded by the observed range of x and ordered within each group. Therefore, cp_i_id_sd is the SD of the normal distribution before truncation rather than the realized SD. See group-level effects with mcp for the model equation and practical interpretation.

The default prior on the SD of group-level change points, e.g., cp_1_id_sd, is \text{Normal}\left(0, \frac{W}{4}\right) T(0, ), \qquad W = \frac{2 (\max(x) - \min(x))}{N + 1} where N is the number of change points. W is the expected width of the interval in which a group’s change point can vary, i.e., between its two neighboring change points. Under the default prior, adjacent change points are on average (\max(x) - \min(x)) / (N + 1) apart. With one change point, W is the observed range of x.

The scale W / 4 is calibrated on the realized spread of the group-level change points rather than on cp_i_id_sd itself. Because group-level change points are truncated to an interval of width W, their realized SD cannot exceed W / \sqrt{12}, the SD of a uniform distribution on that interval. When cp_i_id_sd is larger than about W / 2, the truncated normal is practically uniform, so the data carry little information to distinguish between larger values. With the default prior, the realized SD as a fraction of its maximum W / \sqrt{12} is approximately uniform between 0 and 1 a priori (prior deciles: 0.10, 0.20, 0.30, 0.39, 0.49, 0.59, 0.69, 0.79, 0.87; simulated with \text{cp}_i uniform within the interval). This is similar to how the logit and probit intercept priors are (approximately) uniform on the probability scale (read more). Thus, a priori, group-level change points are exchangeable around the population-level change point, so groups with little information about their change point are pooled towards it.

The prior is half-normal rather than the half-Student-t used for predictor group-level SDs. Because the data carry little information about large values of cp_i_id_sd, a heavy upper tail would pass largely unchanged into the posterior. When adjacent change points both vary by group, the actual interval depends on the neighboring group-level change points, so W is an approximation.

Deviations from rstanarm and brms

Here, a deviation means a default that differs from the precedent that mcp follows for that parameter: rstanarm for intercepts and coefficients and brms for scale parameters. brms’s improper flat coefficient priors are not a precedent, since JAGS does not support them.

mcp deviates from these precedents in a few places:

  • Logit and probit links (binomial(), bernoulli()): mcp uses rstanarm’s autoscaling, but with scales that make the intercept prior (approximately) uniform on the probability scale. For probit, dnorm(0, 1) is exactly uniform on the probability scale. For logit, dnorm(0, 1.5) is approximately so (95% interval from about 0.05 to 0.95; McElreath, 2020, section 11.1). Larger scales, such as rstanarm’s 2.5, place excess prior mass near 0 and 1. This is particularly undesirable in change-point models, where several segments each contribute their own intercept. Group-level SDs use the same scales (1.5 and 1) instead of brms’s 2.5.

  • Log links (poisson(), negbinomial(), gaussian(link = "log")): Coefficients follow rstanarm. The intercept is centered on the data, as in brms, so that it adapts to the magnitude of the counts. But it has normal tails rather than brms’s Student-t tails, because exponentiated Student-t tails imply infinite expected counts (\mathbb{E}[\exp(\eta)] = \infty). Group-level SDs are half-normal for a similar reason: their lighter tails make extreme group multipliers far less likely. With an offset(), the intercept is a log-rate, so mcp computes its location and scale from the observed log-rates, \log(y) - \text{offset}. brms subtracts mean(offset) from the location but computes the scale from the log-counts, so its prior widens when exposures vary, even if the rates do not.

  • Identity link for poisson(): rstanarm applies its unscaled defaults, e.g., normal(0, 2.5) for the intercept, which ignore the magnitude of the counts. mcp uses the Gaussian defaults instead, which are on the response scale.

  • Offsets with identity links: rstanarm centers the intercept on mean(y), ignoring offset(). But with an offset(), the intercept describes y - \text{offset}, so mcp computes its location and scale from y - \text{offset}, just as for log links. brms subtracts mean(offset) from the location but computes the scale from y.

  • Intercept placement: rstanarm and brms place the intercept prior on centered predictors. mcp places the default prior on Intercept_1 at \min(x), which matters when x is far from zero, e.g., calendar years. Likewise, factor contrasts with their own x-terms get their default prior at \min(x), e.g., stateB_1 in y ~ 1 + state * x. When the x-terms are split by factor levels without a common x-term, e.g., y ~ 1 + x:state, the prior on Intercept_1 applies to the average value at \min(x), weighting each level by its share of the data as in \operatorname{mean}(y). User-specified priors on Intercept_1 apply directly at x = 0. Other predictors are not centered, so the prior applies where they are zero. If such a predictor is far from zero, e.g., age in years, center it before fitting or set a prior on Intercept_1. Read more about the mcp model.

  • x-term interactions: rstanarm scales x:z by the SD of the model-matrix column, \operatorname{sd}(x \cdot z), which depends on the location of x and z. For example, it is around 1000 for x:group when x is calendar years. Since x-terms are measured from the change point, the location of x is not meaningful here. mcp instead uses \operatorname{sd}(x) \cdot \operatorname{sd}(z), where 0/1 indicators contribute a factor of 1, so the prior applies per SD of x and z.

  • Factors without an intercept (0 + state): In y ~ 0 + state, R codes a coefficient for every factor level instead of an intercept and contrasts. rstanarm uses its coefficient prior, centered at zero, which is strongly informative when the data are far from zero. This applies where a segment starts at zero: segment 1, and joined segments with no earlier intercept or x-term, e.g., segment 2 in list(y ~ 0, ~ 0 + state). In other joined segments, e.g., ~ 0 + state after y ~ 1 + x, the coefficients are jumps from where the earlier segment left off, so they get the contrast prior.

  • Coefficients in sigma(), shape(), ar(), and ma(), and group-level SDs: The prior scale of slopes is divided by \operatorname{sd}(\text{column}), so that it applies per SD of the predictor, as for mu. Contrasts are not divided because a 0/1 dummy has no units to convert. brms uses flat priors for these coefficients and the same scale for all group-level SDs.

mcp-specific priors

Some parameters have no precedent in rstanarm or brms, so mcp uses its own defaults:

  • Autoregressive and moving-average terms: mcp uses zero-centered dnorm(0, 0.5) T(-1, 1) priors to favor stationary and invertible dynamics, with modest changes across segments and predictors (dnorm(0, 0.25)).

  • Identity links for binomial() and bernoulli(): rstanarm does not support this link, and brms applies its generic defaults (e.g., student_t(3, 0, 2.5) for the intercept), which do not respect that probabilities lie between 0 and 1. mcp uses a uniform intercept (dunif(0, 1)), uniform contrasts on the possible differences (dunif(-1, 1)), and slopes of dnorm(0, 1 / sd(column)).

  • Change points: As described above. The SDs of group-level change points have an mcp-specific prior, described above.

Some articles discuss the default priors in the context of a particular family or model component: