[Python] Bayesian Approach

I have only this problem statement as of now that I will get Annual Values from 2010-2024 (from a reliable source) and monthly values will be derived via proxy (separate model). Since, the monthly values for a particular year will not add up to the Annual value of that year. So i have to devise a method where the monthly values gets adjusted and adds up to the Annual Value.

For this, I did a Bayesian Analysis as below.

Created synthetic Yearly Data from 2010-2024 (Ay);y is 2010-2024 My Assumption is that True_Monthly_Value should be equal to (Ay)/12 Created an observed monthly value and added some noise to it.

# Add noise to the monthly value
np.random.seed(42)
noise = np.random.normal(0,10, size = len(df_monthly))*emphasized text*
df_monthly['Monthly_Observed']= df_monthly['Monthly_Value']+np.abs(noise)


After this I wanted my model to try to remove the noise and estimate something closer to the correct data which is (Ay)/12. I did the following code:

#PyMC model
with pm.Model() as model:

#Setting the PRIOR for Noise from the Monthly_Observed Value
#Because I created the noise thats why this is my PRIOR
#“I believe the noise is positive (via HalfNormal), and probably close to 5, but let the data update that belief.”
sigma = pm.HalfNormal("sigma", sigma=5)


#PRIOR distribution over the latent monthly values
#You're saying:
#“I believe the true values are centered around the noisy monthly_observed values, but they could vary with σ = 10.”
#Monthly_observed is used in this because its the belief that my actual monthly values are closer to the Monthly_Observed values
#with a standard deviation of 10
T_raw = pm.Normal("T_raw", mu=monthly_observed, sigma=10, shape=(n_years, n_months))

#Enforce Constraint i.e. sum of each month in a year = Annual Total
T_total = T_raw.sum(axis=1, keepdims=True)
scaling_factor = annual_total[:, None] / T_total
T = pm.Deterministic("T", T_raw * scaling_factor) ---#check this

# Likelihood: observed noisy monthly values // How probable the observed data is given a specific choice of model parameters
#here’s my noisy observed data monthly_observed. I think these are normally distributed around the true (unknown) monthly values T,
#and the amount of noise in that distribution is described by sigma
M_obs = pm.Normal("M_obs", mu=T, sigma=sigma, observed=monthly_observed)

# Inference
# TRACE comprises the POSTERIOR Samples of the model's parameters
# TUNE is the Burn-in samples
# Burn-in samples are essential for getting accurate Bayesian inference using MCMC
trace = pm.sample(2000, tune=1000, target_accept=0.95, return_inferencedata=True, random_seed=42)

#Extract the de-noised monthly values that the Bayesian Model learned
# Posterior mean of T (shape: n_years x 12)
T_posterior_mean = trace.posterior["T"].mean(dim=("chain", "draw")).values


My Results from the posterior are very good. The r_hat is equivalent to 1.0 and the 96% HDI is also narrow.

Now my doubt is in the model I am using a soft constraint:

#Enforce Constraint i.e. sum of each month in a year = Annual Total
T_total = T_raw.sum(axis=1, keepdims=True)
scaling_factor = annual_total[:, None] / T_total
T = pm.Deterministic("T", T_raw * scaling_factor) ---#check this


I think this is scaling my data and giving me the good results. DO you think its okk to use a soft constraint or I should do something differently. I really need some advise

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