I am trying to do Bayesian analysis using pymc3 and bambi softwares.
My data shape is 136x5 and top 5 rows are as follows:
AGE GENDER AVAR BVAR OUTVAR
0 60 F 0.9 0 8260.0
1 56 F 5.4 1 15888.0
2 55 F 1.2 1 19734.4
3 52 F 1.7 1 15904.2
4 49 F 1.6 0 14848.0
where OUTVAR is target variable and others are predictors.
I used following Python3 code:
from bambi import Model
model = Model(bdf)
results = model.fit('OUTVAR ~ AGE + GENDER + AVAR + BVAR', samples=5000, chains=2)
print(results[1000:].summary())
and got following output:
Auto-assigning NUTS sampler...
Initializing NUTS using advi...
Average Loss = 1,476.2: 21%|██████████████▊ | 10601/50000 [00:01<00:07, 5542.87it/s]
Convergence achieved at 10700
Interrupted at 10,699 [21%]: Average Loss = 1,520.6
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [OUTVAR_sd, BVAR, AVAR, AGE, GENDER, Intercept]
Sampling 2 chains: 100%|██████████████████████████████████████████████████████████████████████████| 11000/11000 [11:40<00:00, 15.71draws/s]
There were 154 divergences after tuning. Increase `target_accept` or reparameterize.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
There were 241 divergences after tuning. Increase `target_accept` or reparameterize.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
The number of effective samples is smaller than 10% for some parameters.
mean sd hpd0.95_lower hpd0.95_upper effective_n gelman_rubin
AGE 6.936836 81.741918 -158.176468 165.050530 589 1.005245
AVAR -78.356403 410.942267 -896.068374 718.650196 2414 1.000314
BVAR 2639.993063 2262.101985 -1528.841953 7297.760056 553 1.000544
GENDER[T.M] -615.092080 1659.905226 -3855.789966 2756.704710 1392 1.003126
Intercept 11739.843222 3591.680335 5239.872556 19053.145349 179 1.007447
OUTVAR_sd 8936.351700 283.640474 8402.347757 9318.495791 6027 1.000028
How do I analyze and what inference can I make from this table of values?
Edit: To be specific, I want to confirm if mean indicates coefficient values for each of variable. However, in usual linear regression, there is no coefficient for outcome variable, while here it is listed. Also, I am not clear what is the meaning of effective_n value for each variable. Thanks for your help.
In this context, mean is the mean of the posterior distribution. For a simple case like this, the posterior means will usually be close to the point estimates you would get from traditional least-squares regression.
The OUTVAR_sd row is not for the outcome variable; it represents the model error. Think of the true OUTVAR score as being the sum of the model-predicted OUTVAR, plus a normally distributed error. Then OUTVAR_sd is the posterior estimate for that error term.
effective_n is a heuristic estimate of the number of independent samples in the chain for each variable. This is almost invariably smaller than the nominal number, because consecutive samples are correlated. See this blog post for a brief explanation.