I am using an insurance claims dataset which only shows claims over a certain size (it is truncated). However, **it also includes the sum of all claims seen** (but not the number). It strikes me that it would be super useful information to help reconstruct the whole distribution.

However, I’m not sure how to integrate this information when fitting my `TruncatedNormal`

. Does anyone have any insight?

Data setup:

```
import numpy as np
import pymc as pm
import matplotlib.pyplot as plt
import arviz as az
print(f"pymc: {pm.__version__}")
rng = np.random.default_rng(20240802)
# insurance losses
draws = rng.lognormal(10, 3, size=100_000)
# total loss - this number is provided
total_loss = draws.sum()
# we only get individual losses reported above this number
min_reported_loss = 1_000_000
print(f"We see only {(draws > min_reported_loss).mean():%} of losses")
observed_losses = draws[draws > min_reported_loss]
print(f"Sampled mean {np.log(draws).mean():.3f}")
print(f"Sampled std {np.log(draws).std():.3f}")
plt.hist(np.log(draws), alpha=0.5)
plt.axvline(np.log(min_reported_loss), color="black", linestyle="--")
```

And the model:

```
print(f"Observed count: {len(observed_losses):,}\n")
with pm.Model() as model:
mu = pm.Normal("mu", 0, 10)
sigma = pm.HalfNormal("sigma", 5)
y = pm.TruncatedNormal(
"y",
mu=mu,
sigma=sigma,
lower=np.log(min_reported_loss),
observed=np.log(observed_losses),
)
with model:
trace = pm.sample()
pm.summary(trace)
```