Convergence issues when running model with categorical covariates in hierarchy

@AlexanderFengler is already working on this.

Since Jun 14, 2024.

Synthetic data:
```
# Generated dataset
n_subjects = 25 # number of subjects
n_trials = 50 # number of trials per subject - vary from low to high values to check shrinkage
sd_v = 0.3 # sd for v-intercept (also used for a)
mean_v = 1.25 # mean for v-intercept
mean_vx = 0.8 # mean for slope of x onto v
mean_vy = 0.2 # mean for slope of x onto v
sd_tz=0.1
mean_a = 1.5
mean_t = 0.5
mean_z = 0.5
data_list = []
param_list =[]
for i in range(n_subjects):
# Make parameters for subject i
intercept = np.random.normal(mean_v, sd_v, size=1)
x = np.random.uniform(-1, 1, size=n_trials)
y = np.random.uniform(-1, 1, size=n_trials)
v_x = np.random.normal(mean_vx, sd_v, size=1)
v_y = np.random.normal(mean_vy, sd_v, size=1)
v = intercept + (v_x * x) + (v_y * y)
a = np.random.normal(mean_a, sd_v, size=1)
z = np.random.normal(mean_z, sd_tz, size=1)
t = np.random.normal(mean_t, sd_tz, size=1)
# v is a vector which differs over trials by x and y, so we have different v for every trial - other params are same for all trials
true_values = np.column_stack(
[v, np.repeat(a, axis=0, repeats=n_trials), np.repeat(z, axis=0, repeats=n_trials), np.repeat(t, axis=0, repeats=n_trials)]
)
# Simulate data
obs_ddm_reg_v = hssm.simulate_data(model="ddm", theta=true_values, size=1)
# store ground truth params
param_list.append(
pd.DataFrame(
{
"intercept": intercept,
"v_x": v_x,
"v_y": v_y,
"a": a,
"z": z,
"t": t,
}
)
)
# Append simulated data to list
data_list.append(
pd.DataFrame(
{
"rt": obs_ddm_reg_v["rt"],
"response": obs_ddm_reg_v["response"],
"x": x,
"y": y,
"subject": i,
}
)
)
# Make single dataframe out of subject-wise datasets
dataset_reg_v_hier = pd.concat(data_list)
dataset_reg_v_hier
```

Model
```
model_reg_v_ddm_hier1A = hssm.HSSM(
data=dataset_reg_v_hier,
# loglik_kind="approx_differentiable",
prior_settings = "safe",
include=[
{
"name": "v",
"formula": "v ~ 1 + x + y + (1 + x + y| subject)",
"prior": {
"Intercept": {"name": "Normal", "mu": 1, "sigma": 2, "initval": 1},
"x": {"name": "Normal", "mu": 0, "sigma": 1, "initval": 0},
"y": {"name": "Normal", "mu": 0, "sigma": 1, "initval": 0},
"1|subject": {"name": "Normal",
"mu": 0, # using non-centered approach so mu's of indiv subject offsets should be 0
"sigma": {"name": "HalfNormal",
"sigma": 1
}, "initval": 0.5
},
"x|subject": {"name": "Normal",
"mu": 0,
"sigma": {"name": "HalfNormal",
"sigma": 0.5,
}, "initval": 0.5
},
"y|subject": {"name": "Normal",
"mu": 0,
"sigma": {"name": "HalfNormal",
"sigma": 0.5,
}, "initval": 0.5
},
},
"link": "identity",
},
{
"name": "t",
"formula": "t ~ 1 + (1 | subject)",
"prior": {
"Intercept": {"name": "Normal", "mu": 0.5, "sigma": 0.4, "initval": 0.3},
"1|subject": {"name": "Normal",
"mu": 0,
"sigma": {"name": "HalfNormal",
"sigma": 0.5, "initval": .1
},
},
},
"link": "identity",
},
{
"name": "z",
"formula": "z ~ 1 + (1 | subject)",
"prior": {
# "Intercept": {"name": "HalfNormal", "sigma": 1, "initval": .5},
"1|subject": {"name": "Normal",
"mu": 0,
"sigma": {"name": "HalfNormal",
"sigma": 0.05, "initval": .01
},
},
},
},
{
"name": "a",
"formula": "a ~ 1 + (1 | subject)",
"prior": {
"Intercept": {"name": "Gamma", "mu": 0.5, "sigma": 1.75, "initval": 1},
"1|subject": {"name": "Normal",
"mu": 0,
"sigma": {"name": "HalfNormal",
"sigma": 1, "initval": 0.3
},
},
},
},
],
noncentered = True,
# p_outlier=0
)
model_reg_v_ddm_hier1A
```

As reported by @frankmj , this samples easily for `analytic`, but somehow gets stuck for `approx_differentiable`.

Need to investigate this.