Bayesian Inference with different types of observations

Questions v5
1

Hello,

I like to build a model, which uses Bayesian Inference to infer the true air speed tas based on observations of the ground speed gs and wind speed wind.

For gs, I can assume it to be Gaussian due to measurement error. Assuming a constant sigma, I’ll get a model for this part like this:

import pymc as pm
import numpy as np

speeds = np.array([190.0, 181.2, 202.5])

with pm.Model() as model:
    mean_gs = pm.Uniform("mean_gs", lower=0, upper=200)
    sigma_gs = 2.0
    gs = pm.Normal("gs", mu=mean_gs, sigma=sigma_gs, observed=speeds)

This is just based on the PyMC tutorial and Bayesian Methods for Hackers. It works as expected. The result is a posterior which represent probable values for mu of the gs Normal distribution.

Now I like to add ws to obtain tas = gs + wind as a distribution. (This is obviously a simplification from a physical standpoint, but ok for the demonstration here.)

I tried it like this:

with pm.Model() as model:
    mean_gs = pm.Uniform("mean_gs", lower=0, upper=200)
    sigma_gs = 5.0
    mean_wind = pm.Uniform("mean_wind", lower=10, upper=40)
    sigma_wind = 2.0
    wind = pm.Normal('wind', mu=mean_wind, sigma=sigma_wind)
    gs = pm.Normal("gs", mu=mean_gs, sigma=sigma_gs, observed=speeds)
    tas = pm.Deterministic('tas', gs + wind)

But then the result are three different posteriors for tas, because I have three observations (will be 5, with 5 observed and so on). If I’ll add observed values to wind, they have to be the same dimension as the observed for gs, otherwise I’ll get an error.

So obviously, this is not the right approach to my problem. However, I couldn’t find any examples for such use cases with contentious distributions and observations from multiple sources. How would I implement such problems with PyMC? Do I need three models, one for gs, one for ws and a third to merge the results of the first two?

Thank you for your thoughts!