# Rounding Error Model

**URL:** <https://discourse.pymc.io/t/rounding-error-model/16391>\
**Category:** version agnostic\
**Created:** [January 16, 2025, 3:27am UTC](https://discourse.pymc.io/t/rounding-error-model/16391 "2025-01-16T03:27:29Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![dreycenfoiles](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/dreycenfoiles/32/8448_2.png) [@dreycenfoiles](https://discourse.pymc.io/u/dreycenfoiles)\
**Post date:** [January 16, 2025, 3:27am UTC](https://discourse.pymc.io/t/rounding-error-model/16391/1 "2025-01-16T03:27:29Z")

</div>

I am working with sensor data that has limited precision which I believe to be important to incorporate into my model. However, I’m struggling with how to implement this. I feel like this is really obvious and I’m just missing something.

I found this section in the Stan user guide: [Measurement Error and Meta-Analysis](https://mc-stan.org/docs/stan-users-guide/measurement-error.html#rounding) but I’m having difficulty converting it to PyMC. They have two versions; the one that I think is closest to PyMC is

```auto
# y = (10, 10, 12, 11, 9)

data {
  int<lower=0> N;
  vector[N] y;
}
parameters {
  real mu;
  real<lower=0> sigma;
  vector<lower=y-0.5, upper=y+0.5>[N] z;
}
model {
  sigma ~ normal(0, 1);
  z ~ normal(mu, sigma);
}

```

trying to convert this to PyMC, I was thinking something like this:

```py
y = np.array([10, 10, 12, 11, 9])

with pm.Model() as rounding_error_model:

    mu = pm.Flat("mu")
    sigma = pm.HalfNormal('sigma', sigma=1)
    z = pm.Normal('z', mu=mu, sigma=sigma, shape=y.shape)

    lower = pm.math.floor(z)
    upper = pm.math.ceil(z)

    y_obs = pm.Uniform('y_obs', observed=y, lower=lower, upper=upper)
    pm.sample()

```

but this won’t sample because the `logp` of `y_obs` will be `-inf`. Are there any ways that I could rework this?

Thank you for your time!

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<div class="post-metadata">

**Author:** ![ricardoV94](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/ricardov94/32/5775_2.png) [@ricardoV94](https://discourse.pymc.io/u/ricardoV94)\
**Post date:** [January 16, 2025, 8:53am UTC](https://discourse.pymc.io/t/rounding-error-model/16391/2 "2025-01-16T08:53:25Z")

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PyMC can give you the probability of a rounded distribution directly. The equivalent to the first model in Stan is something like this (disclaimer: I didn’t run this code):

```python
import pymc as pm

def rounded_normal_dist(mu, sigma, size):
  return pm.math.round(pm.Normal.dist(mu, sigma, size=size))

with pm.Model() as m:
  mu = pm.Flat("mu")
  sigma = pm.HalfNormal("sigma")
  y_obs = pm.CustomDist("y_obs", mu, sigma, dist=rounded_normal_dist, observed=[10, 10, 12, 11, 9])

```

Docstrings on CustomDist, and the `dist` param specifically: [pymc.CustomDist — PyMC v5.13.1 documentation](https://www.pymc.io/projects/docs/en/v5.13.1/api/distributions/generated/pymc.CustomDist.html)

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<div class="post-metadata">

**Author:** ![ricardoV94](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/ricardov94/32/5775_2.png) [@ricardoV94](https://discourse.pymc.io/u/ricardoV94)\
**Post date:** [January 16, 2025, 8:56am UTC](https://discourse.pymc.io/t/rounding-error-model/16391/3 "2025-01-16T08:56:46Z")

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Otherwise, for the second approach is a bit harder to translate. I guess you can do something by specifying an interval transform to z between y±0.5, and you don’t need a likelihood. But I’m not 100% sure on what that model means exactly in Stan.

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<div class="post-metadata">

**Author:** ![dreycenfoiles](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/dreycenfoiles/32/8448_2.png) [@dreycenfoiles](https://discourse.pymc.io/u/dreycenfoiles)\
**Post date:** [January 16, 2025, 3:40pm UTC](https://discourse.pymc.io/t/rounding-error-model/16391/4 "2025-01-16T15:40:16Z")

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Yup… I figured it would be something simple like this. Thank you for your help!

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<div class="post-metadata">

**Author:** ![bob-carpenter](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/bob-carpenter/32/8124_2.png) [@bob-carpenter](https://discourse.pymc.io/u/bob-carpenter)\
**Post date:** [January 20, 2025, 7:16pm UTC](https://discourse.pymc.io/t/rounding-error-model/16391/5 "2025-01-20T19:16:54Z")

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> [@ricardoV94](#):
>
> I’m not 100% sure on what that model means exactly in Stan.

By default, Stan samples uniformly on values that satisfy the constraint, so this

```stan
vector<lower=y-0.5, upper=y+0.5>[N] z;

```

implies the following in more generative terms,

```stan
for (n in 1:N) {
  z[n] ~ uniform(y - 0.5, y + 0.5);
}

```

So it’s just saying the prior is that the value rounded to `y` is uniformly distributed between `y - 0.5` and `y + 0.5`. This is assuming the only noise is measurement error and the rounding is to units.