I need to estimate a probability from a set of successes observed over time. I this application I have multiple ‘looks’ at the data, and need to estimate ‘p’ from this.

Because of this, i thought that a simple Binomial model with saturating growth would be a good place to start:

```
import pandas as pd
import numpy as np
import pymc3 as pm
import matplotlib.pyplot as plt
import seaborn as sns
import theano.tensor as tt
%matplotlib inline
plt.rcParams.update({'font.size': 20})
%config InlineBackend.figure_format = 'retina'
SEED = 48
def logistic(K, r, t, C_0):
A = (K-C_0)/C_0
return K / (1 + A * np.exp(-r * t))
data = {
'y': [1, 20, 30, 55, 80, 95, 105, 110, 111, 112],
'n': [1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000]
}
t = np.linspace(1, 10, 10)
with pm.Model() as binomial_model:
C_0 = pm.Normal("C_0", mu=0.001, sigma=10.0)
r = pm.Normal("r", mu=0.02, sigma=0.5)
K = pm.Normal("K", mu=0.0, sigma=1.0)
# saturating growth
growth = logistic(K, r, t, C_0)
p = pm.Deterministic("p_t", pm.invlogit(growth))
# likelihood
pm.Binomial('y', n=data['n'], p=p, observed=data['y'])
pm.model_to_graphviz(binomial_model)
with binomial_model:
train_trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True)
pm.traceplot(train_trace)
pm.plot_posterior(train_trace)
ppc = pm.sample_posterior_predictive(train_trace)
```

And this model is being fit, and producing reasonable posterior predictions.

```
fig, ax = plt.subplots(figsize=(10, 8))
ax.plot(t, ppc['y'].T, ".k", alpha=0.05)
ax.plot(t, data['y'], color="r")
ax.set(xlabel="Days", ylabel="Converted", title=f"Posterior predictive on the training set")
```

But I’m a bit concerned that it’s producing multi-modal posteriors (even if one mode contains the truth).

What steps could I take to prevent this, and use better priors?