# DNBP Loss BackPropagation

## Colors
- Blue - Indicates Belief Weights Likelihood
- Red - Indicates Message Weights Unary
- Green - Indicates Message Weights Neighbourhood

# Likelihood Optimizer
- This optimizer contains 2 parameters
- CNN network parameters $f_s$
- Node Likelihood parameters $\phi_s$
- **a**
- $\frac {dl(x_i|\theta_1)}{d\phi_s}=\sum^N_{j=1} 1 \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times \frac {dw_{1,i,j}}{d\phi_{s}}$
- **b**
- $\frac {dl(x_i|\theta_1)}{df_s}=\sum^N_{j=1} 1 \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times \frac {dw_{1,i,j}}{dY} \times \frac{dY}{df_s}$
- This optimizer only depends on `Belief Weights Likelihood (loss function)`.
:::info
- **c**
- Backpropagation doesn't happen because `likelihood factors training should depend only on the corresponding node, not its neighbors`. (Comment in the code)
:::

# Sampler Optimizer
- This optimizer contains
- Edge Sampler Parameters $\tilde\psi_{sd}$
- **d**
- $\frac {dl(x_i|\theta_2)}{d\tilde\psi_{sd}}=\sum^N_{j=1} w_{\theta_{2,i,j}} \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times -1 \times \left(\frac {x_i - p_{i,j}}{\sigma} \right) \times \frac {dx_{i}}{d\tilde\psi_{sd}}$
- This optimizer only depends on `Message Weights Unary (loss function)`.

# Density Optimizer
- This optimizer contains
- Edge Density Parameters $\psi_{sd}$
- **e**
- $\frac {dl(x_i|\theta_3)}{d\psi_{sd}}=\sum^N_{j=1} 1 \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times \frac {dw_{3,i,j}}{d\psi_{sd}}$
- This optimizer only depends on `Message Weights Neighbourhood (loss function)`

# Time Optimizer
- This optimizer contains
- Time Sampler Parameters
- **f**
- $\frac {dl(x_i|\theta_1)}{d\tilde\tau_{s}}=\sum^N_{j=1} w_{\theta_{1,i,j}} \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times -1 \times \left(\frac {x_i - p_{i,j}}{\sigma} \right) \times \frac {dx_{i}}{d\tilde\tau_s}$
- **g**
- $\frac {dl(x_i|\theta_2)}{d\tilde\tau_{s}}=\sum^N_{j=1} w_{\theta_{2,i,j}} \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times -1 \times \left(\frac {x_i - p_{i,j}}{\sigma} \right) \times \frac {dx_{i}}{d\tilde\tau_s}$
- **h**
- $\frac {dl(x_i|\theta_3)}{d\tilde\tau_{s}}=\sum^N_{j=1} w_{\theta_{3,i,j}} \times\left[ \frac 1 {\sigma \sqrt {2\pi}} \exp \left( \frac {-1} 2 \times \left(\frac {x_i-p_{i,j}}{\sigma}\right)^2 \right)\right] \times -1 \times \left(\frac {x_i - p_{i,j}}{\sigma} \right) \times \frac {dx_{i}}{d\tilde\tau_s}$
- - This optimizer depends on
- `Belief Weights Likelihood (loss function)`
- `Message Weights Unary (loss function)`
- `Message Weights Neighbourhood (loss function)`