Hello Stan community,

I’ve been trying to troubleshoot a complex Stan model for about 2.5 days, but some sigma fits just go crazy. Here’s a minimal example - can someone tell me what’s wrong with it? The model problem is that we weigh two boxes, twice each, and want estimates for average box weight (mu), standard weight deviation (sigma), and measurement error (m_sd). For the sake of this example, let say that mu=1000g, box1=999g, box2=1001g, sigma = approx 1, and m_sd = appox 1.

Here’s the self-contained Stan model:

```
transformed data {
int<lower=0> N; // total number of measurements
int<lower=0> boxCount; // number of boxes to measure
int<lower=1> B[4]; // index of box that was measured
real<lower=0> M[4]; // weight measurements
boxCount = 2;
N = 4; // 2 measurements per box
B = {1, 1, 2, 2}; // measured box 1 twice, then box 2 twice
M = {999+1, 999-1, 1001-1, 1001+1}; // measured weights
}
parameters {
real<lower=0> W[boxCount]; // true weight of a given box
real<lower=0> mu; // average weight of boxes
real<lower=0> sigma; // standard deviation of box weights
real<lower=0> m_sd; // standard deviation of measurements
}
model {
// True box weights
for (b in 1:boxCount) {
W[b] ~ normal(mu, sigma);
}
// Measured box weights
for (m in 1:N) {
M[m] ~ normal(W[B[m]], m_sd);
}
}
```

The result of 100 iterations is as follows (sigma goes towards infinity with more iterations):

OS: x86_64, darwin13.4.0; rstan: 2.15.1; Rcpp: 0.12.8; inline: 0.3.14

…

Inference for Stan model: temp2.

1 chains, each with iter=100; warmup=50; thin=1;

post-warmup draws per chain=50, total post-warmup draws=50.

`mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat`

W[1] 999.87 18.61 118.90 755.96 957.54 984.58 1044.29 1230.22 41 0.98

W[2] 969.13 23.85 130.73 626.53 931.98 998.59 1016.34 1181.13 30 0.98

mu 1074.46 296.18 1435.16 51.14 421.16 621.86 1065.05 3803.69 23 0.99

sigma 2129.80 819.44 3836.89 501.56 875.56 1005.34 1503.88 15966.03 22 1.03

m_sd 137.75 52.68 134.06 37.96 42.18 83.51 155.66 480.19 6 0.99

lp__ -2.17 1.01 2.38 -6.94 -3.70 -2.18 -0.61 1.84 6 1.12

Why do sigma and m_sd go crazy like that? They should be around 1 or so, not in the hundreds or thousands…

The analysis using linear mixed effects works just fine:

```
dfBox = data.frame(
B = factor(c(1, 1, 2, 2)),
M = c(999+1, 999-1, 1001-1, 1001+1)
)
library(lme4)
lmer1 = lmer(M ~ (1|B), data=dfBox)
> summary(lmer1)
Linear mixed model fit by REML ['lmerMod']
Formula: M ~ (1 | B)
Data: dfBox
REML criterion at convergence: 12.7
Scaled residuals:
Min 1Q Median 3Q Max
-1.0607 -0.5303 0.0000 0.5303 1.0607
Random effects:
Groups Name Variance Std.Dev.
B (Intercept) 1 1.000
Residual 2 1.414
Number of obs: 4, groups: B, 2
Fixed effects:
Estimate Std. Error t value
(Intercept) 1000 1 1000
> fixef(lmer1)
(Intercept)
1000
> ranef(lmer1)
$B
(Intercept)
1 -0.5
2 0.5
> confint(lmer1)
Computing profile confidence intervals ...
2.5 % 97.5 %
.sig01 0.0000000 4.118386
.sigma 0.6739152 3.468133
(Intercept) 997.5865087 1002.413505
```

Thanks in advance!

Ellis