To specify a truncated distribution, you can use the LOWER= and/or UPPER= options. Almost all of the standard distributions, including the GENERAL and DGENERALfunctions, take these optional truncation arguments. The exceptions are the binary and uniform distributions.
For example, you can specify the following:
prior alpha ~ normal(mean = 0, sd = 1, lower = 3, upper = 45);
or
parms beta; a = 3; b = 7; ll = (a + 1) * log(b / beta); prior beta ~ general(ll, upper = b + 17);
The preceding statements state that if beta
is less than b+17
, the log of the prior density is ll
, as calculated by the equation; otherwise, the log of the prior density is missing—the log of zero.
When the same distribution is applied to multiple parameters in a PRIOR statement, the LOWER= and UPPER= truncations apply to all parameters in that statement. For example, the following statements
define a Poisson density for theta
and gamma
:
parms theta gamma; lambda = 7; l1 = theta * log(lambda) - lgamma(1 + theta); l2 = gamma * log(lambda) - lgamma(1 + gamma); ll = l1 + l2; prior theta gamma ~ dgeneral(ll, lower = 1);
The LOWER=1 condition is applied to both theta
and gamma
, meaning that for the assignment to ll
to be meaningful, both theta
and gamma
have to be greater than 1. If either of the parameters is less than 1, the log of the joint prior density becomes a missing
value.
In releases before SAS/STAT 13.1, only three distributions support parameters (or functions of parameters) in the LOWER= and UPPER= options. These are the normal distribution, the GENERAL function, and the DGENERAL function. Appropriate normalizing constants, which are required if the truncations involve model parameters, are not calculated. Starting with SAS/STAT 13.1, PROC MCMC calculates the normalizing constant in all truncated distributions, and you can use parameters in the LOWER= or UPPER= option.
Note that if you use either the GENERAL or DGENERAL function, you must compute the normalizing constant in cases where it is required. A truncated distribution has the probability distribution
where is the density function and is the cumulative distribution function. In SAS functions, is the probability density function and is the cumulative distribution function. The following example shows how to construct a truncated gamma prior on theta
, with SHAPE=3, SCALE=2, LOWER=A, and UPPER=B:
lp = logpdf('gamma', theta, 3, 2) - log(cdf('gamma', a, 3, 2) - cdf('gamma', b, 3, 2)); prior theta ~ general(lp);
This density specification is different from the following more naive definition, without taking into account the normalizing constant:
lp = logpdf('gamma', theta, 3, 2); prior theta ~ general(lp, lower=a, upper=b);
If a
or b
is a parameter, you get very different results from the two formulations.
There is no built-in mechanism in PROC MCMC that models censoring automatically. You need to construct the density function (using a combination of the LOGPDF, LOGCDF, and LOGSDF functions and IF-ELSE statements) for the censored data.
Suppose you partition the data into four categories: uncensored (with observation x
), left censored (with observation xl
), right censored (with observation xr
), and interval censored (with observations xl
and xr
). The likelihood is the normal with mean mu
and standard deviation s
. The following statements construct the corresponding log likelihood for the observed data:
if uncensored then ll = logpdf('normal', x, mu, s); else if leftcensored then ll = logcdf('normal', xl, mu, s); else if rightcensored then ll = logsdf('normal', xr, mu, s); else /* this is the case of interval censored. */ ll = log(cdf('normal', xr, mu, s) - cdf('normal', xl, mu, s)); model general(ll);