# A Guide to Prior Selection

In Bayesian modeling, a **prior distribution** (or simply "prior") represents our beliefs about a parameter *before* we have seen the data. Choosing appropriate priors is a key part of the modeling process. This guide explains the rationale behind the prior choices in the ConversionFlow library.

## The Role of Priors

Priors are a way to incorporate domain knowledge into the model. They can be:

-   **Informative**: A narrow distribution that strongly pulls the parameter towards a certain value.
-   **Weakly Informative**: A wider distribution that gently guides the model, preventing it from exploring completely unrealistic parameter values.
-   **Uninformative (Flat)**: A very wide distribution that provides little to no information to the model.

In ConversionFlow, we primarily use **weakly informative priors**. This is a best practice in modern Bayesian modeling, as it helps to regularize the model (preventing overfitting) without overly biasing the results.

## Priors in `config.yml`

The priors for the `beta` coefficients (the strength of the relationships between nodes) are defined in the `priors.beta_distributions` section of the `config.yml` file.

### `HalfCauchy`

The `HalfCauchy` distribution is often used as a weakly informative prior for scale parameters (parameters that must be non-negative). In our case, we use it for `beta` coefficients where we have a strong prior belief that the effect should be positive.

For example, we expect that `session_start` can only have a positive or zero effect on `download`. A user cannot be *less* likely to download a brochure simply because they started a session.

```yaml
session_start:
  download: { distribution: HalfCauchy, sigma: 5 }
```

The `sigma` parameter controls the width of the distribution. A larger `sigma` allows for larger effects.

### `StudentT`

The `StudentT` distribution is a good choice for a weakly informative prior for regression coefficients. It is similar to the `Normal` distribution but has "heavier tails," which means it is more robust to outliers.

```yaml
car_configuration:
  configdetailsevent: { distribution: StudentT, nu: 3, sigma: 2 }
```

-   `nu`: The degrees of freedom. A lower `nu` results in heavier tails. `nu=3` is a common choice.
-   `sigma`: The scale parameter, similar to the standard deviation in a `Normal` distribution.

### `Normal`

The `Normal` distribution is a good default choice for a weakly informative prior when you believe the effect is likely to be centered around a certain value (usually 0).

```yaml
feature_engagement:
  add_to_cart: { distribution: Normal, mu: 0, sigma: 2.5 }
```

-   `mu`: The mean of the distribution (the center of your prior belief).
-   `sigma`: The standard deviation (how much you expect the parameter to vary).

## Choosing Your Own Priors

When customizing the model, think about the relationships you are modeling:

-   If you have a strong belief that an effect can only be positive, a `HalfCauchy` or `HalfNormal` prior is a good choice.
-   If you believe an effect could be positive or negative, but is likely to be small, a `Normal` or `StudentT` distribution centered at 0 is a good choice.
-   If you have strong domain knowledge that suggests a different center or scale for your prior, you can adjust the `mu` and `sigma` parameters accordingly.