# Finding parameters with different model settings (y = a*x + b*z or x = (1/a)*y - (b/a)*z)

Hi,

I have just a general question.
I would like to know if my assumption holds.

Here, I have three data sets – y, x, and z. They are all independent.

If I want to analyze the relationships among them, I can build a linear model such as:

Model 1)
y = a x + bz

Soon after, I found that y is not normally distributed but log(e^{y}) does.
So we can modify Model 1) to:

Model 2)
log(e^{y}+1) = log(e^{ax+bz}+1)

At he same, I found that x has normality. So, I changed the equation to:

Model 3)
x = {1\over a}y - {b \over a}z

Now, we can use the Normal likelihood function to find a and b.

Such as:

with pm.Model() as model_2
a = pm.Normal('a', mu=0, sigma=1)
b = pm.Normal('b', mu=0, sigma=1)
mu = pm.math.log(pm.math.exp(ax+bz)+1)

yt = np.log(np.exp(y)+1)
y_obs = pm.Normal('obs', mu=mu, sigma=1, observed=yt)
pm.sample()

with pm.Model() as model_3
a = pm.Normal('a', mu=0, sigma=1)
b = pm.Normal('b', mu=0, sigma=1)
mu = 1/a*y - b/a*z

x_obs = pm.Normal('obs', mu=mu, sigma=1, observed=x)
pm.sample()


After the samplings from model_2 and model_3, should we get the same or similar distributions of a and b from the trace plots?