I am trying to do a Bayesian analysis using a model that comes from the literature in non-Bayesian form: $y = \Phi(\frac{1}{\alpha} * log(A/\beta)$. Because the model uses the function $\Phi$ its outcome is in the range $0-1$. My observations cannot exceed this range either (but can be 0 or 1). However, using a Gaussian in the likelihood function means that the uncertainty bounds are sometimes outside the $0-1$ range (see Figure below). Statistically speaking, is this a problem, and if so, what would be a better solution?

$R_i \sim \mathcal{N}(\mu,\sigma)$

$\mu_i = \Phi \Bigg(\frac{1}{\alpha} * log(A/\beta) \Bigg)$

$\alpha \sim \mathcal{N}(0.35, 0.1) $

$\beta\sim \mathcal{N}(70, 10) $

$\sigma \sim \mathcal{U}(0, 0.3) $

I am trying to do a Bayesian analysis using a model that comes from the literature in non-Bayesian form: $y = \Phi\Bigg(\frac{1}{\alpha} * log(A/\beta)\Bigg)$. Because the model uses the function $\Phi$ its outcome is in the range $0-1$. My observations cannot exceed this range either (but can be 0 or 1). However, using a Gaussian in the likelihood function means that the uncertainty bounds are sometimes outside the $0-1$ range (see Figure below). Statistically speaking, is this a problem, and if so, what would be a better solution?

$R_i \sim \mathcal{N}(\mu,\sigma)$

$\mu_i = \Phi \Bigg(\frac{1}{\alpha} * log(A/\beta) \Bigg)$

$\alpha \sim \mathcal{N}(0.35, 0.1) $

$\beta\sim \mathcal{N}(70, 10) $

$\sigma \sim \mathcal{U}(0, 0.3) $

I am trying to do a Bayesian analysis using a model that comes from the literature in non-Bayesian form: $y = \Phi(\frac{1}{\alpha} * log(A/\beta)$. Because the model uses the function $\Phi$, the outcome of the model is in the range $0-1$. My observations cannot exceed this range either (but can be 0 or 1). However, using a Gaussian in the likelihood function means that the uncertainty bounds are sometimes outside the $0-1$ range (see Figure below). Statistically speaking, is this a problem, and if so, what would be a better solution?

$R_i \sim \mathcal{N}(\mu,\sigma)$

$\mu_i = \Phi \Bigg(\frac{1}{\alpha} * log(A/\beta) \Bigg)$

$\alpha \sim \mathcal{N}(0.35, 0.1) $

$\beta\sim \mathcal{N}(70, 10) $

$\sigma \sim \mathcal{U}(0, 0.3) $

I am trying to do a Bayesian analysis using a model that comes from the literature in non-Bayesian form: $y = \Phi(\frac{1}{\alpha} * log(A/\beta)$. Because the model uses the function $\Phi$ its outcome is in the range $0-1$. My observations cannot exceed this range either (but can be 0 or 1). However, using a Gaussian in the likelihood function means that the uncertainty bounds are sometimes outside the $0-1$ range (see Figure below). Statistically speaking, is this a problem, and if so, what would be a better solution?

$R_i \sim \mathcal{N}(\mu,\sigma)$

$\mu_i = \Phi \Bigg(\frac{1}{\alpha} * log(A/\beta) \Bigg)$

$\alpha \sim \mathcal{N}(0.35, 0.1) $

$\beta\sim \mathcal{N}(70, 10) $

$\sigma \sim \mathcal{U}(0, 0.3) $