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EDIT3: The model with double exponential prior (Lasso) gaves me bigger Deviance, BIC and DIC values than the model with Gaussian priors and I even get a smaller values after removing the dispersion coefficient $\delta_2$ in the Gaussian model.

EDIT2: I fitted two models, one with Gaussian priori for $\beta_j$, $\delta_j$ and one with Laplace(double-exponential).

The estimatives for the Gaussian model are

            Mean      SD  Naive SE Time-series SE
B[1]     -1.17767 0.07112 0.0007497      0.0007498
B[2]     -0.15624 0.03916 0.0004128      0.0004249
B[3]      0.15600 0.05500 0.0005797      0.0005889
B[4]      0.07682 0.04720 0.0004975      0.0005209
delta[1] -3.42286 0.32934 0.0034715      0.0034712
delta[2]  0.06329 0.27480 0.0028966      0.0028969
delta[3]  1.06856 0.34547 0.0036416      0.0036202
delta[4] -0.32392 0.26944 0.0028401      0.0028138

The estimatives for the Lasso model are

              Mean      SD  Naive SE Time-series SE
B[1]     -1.143644 0.07040 0.0007421      0.0007422
B[2]     -0.160541 0.05341 0.0005630      0.0005631
B[3]      0.137026 0.05642 0.0005947      0.0005897
B[4]      0.046538 0.04770 0.0005028      0.0005134
delta[1] -3.569151 0.27840 0.0029346      0.0029575
delta[2] -0.004544 0.15920 0.0016781      0.0016786
delta[3]  0.411220 0.33422 0.0035230      0.0035629
delta[4] -0.034870 0.16225 0.0017103      0.0017103
lambda    7.269359 5.45714 0.0575233      0.0592808

The estimatives for $\delta_2$ and $\delta_4$ reduced a lot in Lasso model, it means that I should remove this variables from the model?

I have a dataset with three variables, where all variables are quantitatives. Let call it $y$, $x_1$ and $x_2$. I'm fitting a regression model in a Bayesian perspective via MCMC with rjags

I done a exploratory analysis and the scatterplot of $y\times x_2$ suggest that a quadratic term should be used. Then I fitted two models

(1) $y=\beta_0+\beta_1*x_1+\beta_2*x_2$

(2) $y=\beta_0+\beta_1*x1+\beta_2*x_2+\beta_3*x_1x_2+\beta_4*x_1^2+\beta_5*x_2^2$

In model 1 the effect size of each parameter is not small and the 95% credible interval not contains the value $0$.

In model 2 the effect size of parameters $\beta_3$ and $\beta_4$ are small and each of credible intervals for all parameters contains $0$.

The fact that a credible interval contains $0$ is enough to say that the parameter is not significant?

Then I adjusted the following model

(3)$y=\beta_0+\beta_1*x_1+\beta_2*x_2+\beta_3*x^2_2$

The effect size of each parameter is not small, but with exception of $\beta_1$ all credible intervals contains $0$.

Which is the right way to do variable selection in Bayesian statistics?

**EDIT:**Below is a example of a parameter which credible interval contains the value 0.

In this case is reasonable say that $\beta_4\neq 0$?

I have a dataset with three variables, where all variables are quantitatives. Let call it $y$, $x_1$ and $x_2$. I'm fitting a regression model in a Bayesian perspective via MCMC with rjags

I done a exploratory analysis and the scatterplot of $y\times x_2$ suggest that a quadratic term should be used. Then I fitted two models

(1) $y=\beta_0+\beta_1*x_1+\beta_2*x_2$

(2) $y=\beta_0+\beta_1*x1+\beta_2*x_2+\beta_3*x_1x_2+\beta_4*x_1^2+\beta_5*x_2^2$

In model 1 the effect size of each parameter is not small and the 95% credible interval not contains the value $0$.

In model 2 the effect size of parameters $\beta_3$ and $\beta_4$ are small and each of credible intervals for all parameters contains $0$.

The fact that a credible interval contains $0$ is enough to say that the parameter is not significant?

Then I adjusted the following model

(3)$y=\beta_0+\beta_1*x_1+\beta_2*x_2+\beta_3*x^2_2$

The effect size of each parameter is not small, but with exception of $\beta_1$ all credible intervals contains $0$.

Which is the right way to do variable selection in Bayesian statistics?

EDIT: I can use Lasso in any regression model, like Beta model? I'm using a model with variable dispersion where $$log(\sigma)=-\pmb{\delta}X$$ where $\pmb{\delta}$ is a vector. I should use Laplace prior in $\pmb{\delta}$ too?