# Are transformed parameters included in the search space for optimizations?

Here is a simplified version of the regression model that I’m working on:

``````data {
int N;
vector[N] x;
vector[N] y;
}
parameters {
real<lower=0> shape;
real<lower=0> a;
real<lower=0> b;
}
transformed parameters {
vector[N] mu = a + b * x;
vector[N] rate = shape ./ mu;
}
model {
y ~ gamma(shape, rate);
}
``````

Below is the rstan code. First, generate fake data:

``````set.seed(1)
a <- 3
b <- 7
shape <- 20
x <- runif(1000, min = 0, max = 10)
mu <- a + b * x
rate <- shape / mu
y <- rgamma(1000, shape = shape, rate = rate)
``````

Use Stan to get maximum-likelihood estimates of parameters:

``````stanmodel <- stan_model('stan_discourse_question.stan')
standata <- list(N = length(x), x = x, y = y)
mle <- optimizing(stanmodel, data = standata)
``````

Print out the results:

``````str(mle)
# List of 3
# \$ par        : Named num [1:2003] 18.48 2.92 6.98 21.44 28.88 ...
# ..- attr(*, "names")= chr [1:2003] "shape" "a" "b" "mu" ...
# \$ value      : num -3367
# \$ return_code: int 0

mle\$par[1:20]
# shape         a         b     mu     mu     mu     mu     mu     mu     mu     mu     mu    mu    mu
# 18.476546  2.915327  6.976342 21.438120 28.875964 42.879538 66.275011 16.985349 65.590066 68.819107 49.014843 46.804476  7.225748 17.284818
# mu    mu    mu    mu    mu    mu
# 15.232530 50.844392 29.711717 56.622099 37.636530 52.978850
``````

The estimated parameters include the estimates for the vectors `mu` and `rate`. Are the `mu` and `rate` vectors included in the search space for the optimization? I suspect that they’re not, since the transformed parameters are simple functions of the parameters and the data, but I want to double check that the search space for my model is not substantially increased by including transformed parameters.

No

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