Looking for help on a few Jacobians

Modeling
2

I’ll give it a try. Someone should check my math below. However, on 1, since you have 1 function but 4 parameters the jacobian is 1 x 4 which has no determinant. You can put the prior on which acts as a weighting function in the high density region of the prior. See for more Once more about Stan and Jacobian warnings | Models Of Reality

For 2,

\begin{aligned} m &= \frac{\sqrt{n} \sum_{i=1}^{n} \frac{y_i - \mu}{\sigma}}{n \sigma} \\ &= \frac{\sum_{i=1}^{n} (y_i - \mu) }{\sqrt{n} \sigma^2} \\ \frac{\partial m}{\partial y} &= \frac{1}{\sqrt{n}\sigma^2} \end{aligned}

the log abs of which is - (0.5 \log(n) + 2 \log(\sigma)). You can drop the first part since it’s constant so you just need

target += -2 * log(sigma);

For the second part, check my math.

\begin{aligned} r &= \frac{y - \mu}{\sigma} \\ v &= \frac{1}{n} \sum_{i=1}^{n} (r - \mu_r)^2 \\ \frac{\partial v}{\partial y} &= \frac{2 \sum_{i=1}^{n} (r - \mu_r) }{n \sigma} \\ &= 0 \end{aligned}

update Thinking about this more I think that the second part doesn’t need an adjustment because \mu_r is 0 and then \sum r / n is also 0. It just boils down to

y <- rnorm(1000, mean = 3, sd = 2)
mike_mod <- cmdstan_model("mike_jacobian.stan")
fit <- mike_mod$sample(
  data = list(n = length(y),
              y = y),
  parallel_chains = 2,
  chains = 2,
  seed = 12312,
  adapt_delta = 0.8,
  max_treedepth = 10,
  iter_sampling = 500,
  iter_warmup = 500
)
fit$summary()

where the Stan code is

data{
  int n;
  vector[n] y ;
}
transformed data{
  real sqrt_n = sqrt(n) ;
  real var_gamma_par = (n-1)/2 ;
}
parameters{
  real mu ;
  real<lower=0> sigma ;
}
model{
  vector[n] resid = (y-mu)/sigma ; // computation involves data and multiple parameters 
  real m = mean(resid) / (sigma/sqrt_n) ;
  real v = variance(resid) ;
  
  m ~ std_normal() ; // needs jacobian?
  v ~ gamma(var_gamma_par,var_gamma_par) ;
  target += -2 * log(sigma);
}

The output as

Both chains finished successfully.
Mean chain execution time: 0.3 seconds.
Total execution time: 0.5 seconds.
> fit$summary()
# A tibble: 3 x 10
  variable    mean  median     sd    mad      q5     q95  rhat ess_bulk ess_tail
  <chr>      <dbl>   <dbl>  <dbl>  <dbl>   <dbl>   <dbl> <dbl>    <dbl>    <dbl>
1 lp__     -501.   -500.   0.983  0.661  -503.   -500.    1.00     468.     667.
2 mu          2.97    2.97 0.128  0.127     2.77    3.18  1.00     935.     712.
3 sigma       2.01    2.01 0.0445 0.0457    1.94    2.09  1.00     692.     681.
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