The main focus here is to explain the default priors used in
bayesnec and to showcase how the user can interrogate the
priors used in a bayesec model and alternatively specify
their own priors, should they wish to. This might be needed depending on
the model and the data because bayesnec tries to find
reasonable and yet only weakly informative priors for the model
parameters by default. First we describe the default priors used in
bayesnec and then follow up with a demonstration of how the
user can specify priors in multiple ways for objects of class
bayesnecfit and bayesmanecfit.
bayesnecThe default priors used in bayesnec can generally be
considered “weakly informative”. They are constructed for each parameter
of each model being fitted based on the characteristics of either the
input response and predictor data, depending on which is relevant to the
specific parameter scaling. In the case of parameters that scale with
the response, priors are constructed based on the relevant link scaling,
whether that be identity or the default (or user specific) link function
for a specific distribution. The priors are constructed by
bnec internally based on the chosen model, the distribution
(including the relevant link function), the predictor and the
response.
Only the parameters top and bot
scale specifically with the response. For Gaussian-distributed response
data (or any response data for which the link ensures valid values of
the response can take from +Inf to -Inf,
including log and logit) priors are
normal with a standard deviation of 2.5 and a
mean set at the 90th and 10th quantiles for
top and bot respectively. For
Poisson-, Negative Binomial- and Gamma-distributed data the response is
bounded by 0 and thus priors are Gamma, with a
mean scaled to correspond to the 75th and 25th
quantiles for top and bot
respectively. The mean is linked mathematically to the shape
(s) and rate parameters (r) by the equation \(mean = s \times (1/r)\) with the Gamma
shape parameter set at 2. For the Binomial, Beta, and Beta Binomial
families estimates for top and bot
must necessarily be constrained between 0 and
1 when modelled on the "identity" link (the
default in bayesnec). Because of this constraint there is
no need to adjust scaling based on the response. bayesnec
uses beta(5, 1) and beta(1, 5) priors to
provide a broad density centred across the upper and lower
0 to 1 range for the top and
bot parameters, respectively.
The parameters nec and ec50 scale
with respect to the predictor because both of these are parameters in
concentration-response curves the are estimated in units of
concentration. To stabilise model fitting the nec and
ec50 parameters are bounded to the upper and lower
observed range in the predictor, under the assumption that the range of
concentrations in the experiment were sufficient to cover the full range
of the response outcomes. The priors used reflect the characteristics of
the observed data that are used to guess the appropriate distribution.
If the predictor data are bounded to 0 and
>1, a Gamma prior is used, with maximum
density (mean, see above) at the median value of the predictor, and a
shape parameter of 5. If the predictor data are bounded to
0 and 1 a beta(2, 2) prior is
used. For predictor data ranging from +Inf to
-Inf a normal prior is used, with a mean set
at the median of the predictor values and a standard deviation of
2.5.
For the parameters beta, slope,
d and f, we first ensured any relevant
transformations in the model formula such that theoretical values of
-Inf to +Inf are allowable, and a
normal(0, 1) prior is used. For example, in the
nec3param model, beta is an
exponential decay parameter which must by definition be bounded to
0 and +Inf. Calling exp(beta) in
the model formula ensures the exponent meets these requirements. Note
also that a mean of 0 and standard deviation of
1 represents a relatively broad prior on the exponential
scaling. See the Model details vignette or
?model("all") for more information on all the models
available in bayesnec and their specific formulation.
There may be situations were the default bayesnec priors
to not behave as desired, or the user wants to provide informative
priors. For example the default priors may be too informative, yielding
unreasonably tight confidence bands, although this is only likely where
there are few data. Conversely, priors may be too vague, leading to poor
model convergence. Alternatively, as indicated in the example below, the
default priors may be of the wrong statistical distribution if there was
insufficient information in the provided data for bayesnec
to guess correctly the appropriate ones to use.
The priors used in the default model fit can be extracted using
pull_prior, and a sample or plot of prior values can be
obtained from the individual brms model fits through the
function sample_priors which samples directly from the
prior element in the brm model fit. We can
also use the function check_prior (based on the
hypothesis function of brms) to assess how the
posterior probability density for each parameter differs from that of
the prior.
To set specified priors, it is simplest to start by letting
bnec find the priors on its own, i.e. by not specifying the
brm argument prior at all.
library(brms)
library(bayesnec)
data(nec_data)
# a single model
exmp_a <- bnec(y ~ crf(x, model = "nec3param"), data = nec_data,
family = Beta(link = "identity"),
iter = 1e4, control = list(adapt_delta = 0.99))class(exmp_a)We can view the prior and posterior probability densities of all the
parameters in the model using the function check_prior,
based on the hypothesis function of brms. This
can be useful to assess if priors are suitably vague, and/or if they
might be having an undesirable influence on the posterior.
check_priors(exmp_a)In this case the priors seem reasonably vague, however there will be
times when it is necessary to modify these priors. The user can take
advantage of the function pull_prior to inspect what
bnec came up with on its own, and decide how best to modify
those priors to be more desirable.
pull_prior(exmp_a)bnec chose a gamma prior on the
NEC parameter of nec3param because the
predictor nec_data$x is non-zero positive. However, imagine
that in theory the predictor could have had negative values, it just
happened to not have in this particular dataset. So let’s go ahead and
specify something else, say a normal with larger variance.
my_prior <- c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("gamma(0.5, 2)", nlpar = "beta"))
exmp_b <- bnec(y ~ crf(x, model = "nec3param"), data = nec_data,
family = Beta(link = "identity"), prior = my_prior,
iter = 1e4, control = list(adapt_delta = 0.99))Two things are of note. If the user is specifying their own priors,
bnec requires them to specify priors for
all parameters. The pull_prior function
shows the priors after the model was fitted, but suppose the
user does not know what parameters were comprised in a particular model.
In those instances, the user can call the function
show_params(model = "all") to inspect the parameters of
each function, or some targeted function in particular.
show_params(model = "nec3param")The user can also specify a named list of priors when one or more models are being fitted to the same dataset.
my_priors <- list(nec3param = c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("gamma(0.5, 2)", nlpar = "beta")),
nec4param = c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("gamma(0.5, 2)", nlpar = "beta"),
prior_string("beta(1, 5)", nlpar = "bot")))
exmp_c <- bnec(y ~ crf(x, model = c("nec3param", "nec4param")), data = nec_data,
family = Beta(link = "identity"), prior = my_priors,
iter = 1e4, control = list(adapt_delta = 0.99))pull_prior also works for an object of class
bayesmanecfit, as does check_priors which
allows an option of passing a filename to save the prior and posterior
probability density plots to a pdf.
pull_prior(exmp_c)
check_priors(exmp_c, filename = "Check_priors")The user can also specify priors for one model only out of the entire
set, bayesnec will return a message stating that it is
searching for priors on its own when they are either ill-formed
(e.g. incomplete or have a typo), or the user simply decided not to
specify priors for a particular model, e.g.
my_priors <- list(nec3param = c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("gamma(0.5, 2)", nlpar = "beta")),
nec4param = c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("gamma(0.5, 2)", nlpar = "beta"),
prior_string("beta(1, 5)", nlpar = "bot")))
exmp_d <- bnec(y ~ crf(x, model = c("nec3param", "nec4param")), data = nec_data,
family = Beta(link = "identity"), prior = my_priors[1],
iter = 1e4, control = list(adapt_delta = 0.99))prior = my_priors[[1]] would also have worked because
the argument priors can either take a brmsprior object
directly, or a named list containing model-specific
brmsprior objects.
Finally the user can also extend an existing
bayesmanecfit object with the function amend,
also by specifying custom-built priors.
necsigm_priors <- c(prior_string("beta(5, 1)", nlpar = "top"),
prior_string("gamma(2, 6.5)", nlpar = "beta"),
prior_string("normal(1.3, 2.7)", nlpar = "nec"),
prior_string("normal(0, 2)", nlpar = "d"))
exmp_e <- amend(exmp_d, add = "necsigm", priors = necsigm_priors)