Hi,

I’m building a logistic regressor to use on survey data where 80%+ of the data may be zeros. I’m running into issues sampling even basic models and wondering whether there are some methods to transform the data or specify the model differently so as to facilitate sampling.

All data is boolean cast as int. the relevant features of dataset are described below:

```
q1_0 q2_0 q3_0
count 100 100 100
mean 0.22 0.68 0.95
std 0.41 0.46 0.21
min 0 0 0
25% 0 0 1
50% 0 1 1
75% 0 1 1
max 1 1 1
```

I’m testing on random data where:

`y = q1_0 + 2 * q2_0 + -2 * q3_0`

The model is:

```
def bernoulli(X,y):
n_features = X.shape.eval()[1]
with pm.Model() as model:
mu_alpha = pm.Normal('mu_alpha', mu=0, sd=100)
sigma_alpha = pm.HalfNormal('sigma_alpha', sd=100)
mu_beta = pm.Normal('mu_beta', mu=0, sd=100)
sigma_beta = pm.HalfNormal('sigma_beta', sd=100)
alpha = pm.Normal('alpha', mu=mu_alpha, sd=sigma_alpha)
beta = pm.Normal('beta', mu=mu_beta, sd=sigma_beta, shape=n_features)
p = pm.invlogit(alpha + _dot(X,beta))
y_obs = pm.Bernoulli('y_obs', p, observed=y)
return model
```

During sampling I got 52 divergences:

```
Sampling 4 chains, 52 divergences: 100%|██████████| 6000/6000 [03:31<00:00, 28.34draws/s]
There was 1 divergence after tuning. Increase `target_accept` or reparameterize.
The acceptance probability does not match the target. It is 0.9978061165399938, but should be close to 0.8. Try to increase the number of tuning steps.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
There was 1 divergence after tuning. Increase `target_accept` or reparameterize.
The acceptance probability does not match the target. It is 0.996341172826582, but should be close to 0.8. Try to increase the number of tuning steps.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
There were 26 divergences after tuning. Increase `target_accept` or reparameterize.
The acceptance probability does not match the target. It is 0.9689980122205897, but should be close to 0.8. Try to increase the number of tuning steps.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
There were 24 divergences after tuning. Increase `target_accept` or reparameterize.
The acceptance probability does not match the target. It is 0.9708234101690537, but should be close to 0.8. Try to increase the number of tuning steps.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
The rhat statistic is larger than 1.4 for some parameters. The sampler did not converge.
The estimated number of effective samples is smaller than 200 for some parameters.
```

summary:

```
mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mu_alpha 0.239 45.384 -94.697 91.707 0.881 2.162 2656.0 221.0 1433.0 1429.0 1.21
mu_beta -0.022 0.081 -0.218 0.086 0.036 0.027 5.0 5.0 6.0 13.0 2.67
alpha -1.649 0.334 -2.252 -0.981 0.082 0.059 17.0 17.0 17.0 28.0 1.16
beta[0] 0.017 0.101 -0.189 0.267 0.033 0.024 9.0 9.0 10.0 28.0 2.13
beta[1] -0.027 0.123 -0.398 0.151 0.051 0.038 6.0 6.0 8.0 12.0 2.56
beta[2] -0.024 0.102 -0.246 0.155 0.032 0.023 10.0 10.0 14.0 19.0 1.82
beta[3] -0.034 0.097 -0.262 0.101 0.031 0.022 10.0 10.0 14.0 17.0 1.87
beta[4] -0.037 0.085 -0.232 0.058 0.024 0.017 13.0 13.0 18.0 56.0 1.81
beta[5] -0.012 0.134 -0.353 0.236 0.047 0.035 8.0 8.0 9.0 15.0 2.37
beta[6] -0.016 0.108 -0.303 0.159 0.021 0.026 26.0 9.0 24.0 25.0 2.24
beta[7] 0.037 0.132 -0.182 0.335 0.027 0.024 24.0 16.0 26.0 27.0 2.26
beta[8] -0.025 0.088 -0.248 0.104 0.029 0.021 9.0 9.0 9.0 23.0 1.84
beta[9] -0.044 0.129 -0.400 0.117 0.052 0.039 6.0 6.0 11.0 13.0 1.95
beta[10] -0.040 0.135 -0.358 0.166 0.043 0.032 10.0 10.0 14.0 17.0 2.03
beta[11] -0.001 0.084 -0.203 0.157 0.017 0.015 24.0 17.0 19.0 46.0 2.20
beta[12] -0.024 0.098 -0.280 0.141 0.039 0.029 6.0 6.0 9.0 14.0 2.19
beta[13] -0.038 0.153 -0.263 0.158 0.048 0.035 10.0 10.0 9.0 14.0 2.11
beta[14] 0.050 0.150 -0.216 0.383 0.034 0.032 20.0 11.0 18.0 17.0 2.35
beta[15] -0.012 0.082 -0.206 0.126 0.014 0.014 36.0 18.0 34.0 28.0 2.16
beta[16] -0.032 0.161 -0.399 0.287 0.066 0.049 6.0 6.0 8.0 11.0 2.64
beta[17] -0.012 0.118 -0.278 0.214 0.035 0.025 11.0 11.0 10.0 34.0 2.24
beta[18] -0.035 0.123 -0.332 0.164 0.048 0.035 7.0 7.0 10.0 14.0 2.13
beta[19] -0.020 0.120 -0.229 0.232 0.032 0.023 14.0 14.0 15.0 24.0 2.30
beta[20] -0.035 0.121 -0.275 0.137 0.042 0.031 8.0 8.0 10.0 15.0 2.00
beta[21] -0.055 0.140 -0.428 0.121 0.059 0.044 6.0 6.0 9.0 13.0 2.02
sigma_alpha 58.925 50.717 0.044 151.255 6.262 4.448 66.0 66.0 45.0 655.0 1.06
sigma_beta 0.061 0.090 0.000 0.225 0.036 0.027 6.0 6.0 5.0 11.0 2.53
```

my sense is that the predominance of zeros and the varying frequency of ones in the relevant features is making it difficult to model. Any ideas what I can do to improve sampling?