This vignette will present how to generate predictions using `estimate`

. Warning: we will go **full Bayesian**. If you’re not familiar with the Bayesian framework, we recommend starting with **this gentle introduction**.

Generating prediction from the model can be used for a wide variety of reasons, one of them being visualisation. This can be achieved via the `estimate_response()`

function and its visualisation spinoff, `estimate_link()`

.

Let’s start by fitting a Bayesian linear regression.

We might be interested in comparing the values predicted by the model to the actual “true” values. This can be done by generating predictions:

```
library(modelbased)
predicted <- estimate_response(model)
head(predicted)
```

```
> Sepal.Length Predicted SE CI_low CI_high
> 1 5.1 2.4 0.87 0.62 4.1
> 2 4.9 2.0 0.87 0.17 3.6
> 3 4.7 1.6 0.90 -0.12 3.4
> 4 4.6 1.5 0.87 -0.35 3.2
> 5 5.0 2.2 0.88 0.52 3.9
> 6 5.4 2.9 0.89 1.19 4.7
```

The output is a data frame containing predicted values (the median and CI of the posterior distribution) for **each of the value of the original data frame** (used for fitting the model). Hence, we can simply add the prediction column (`Predicted`

) to the original dataset and plot the original against the predicted data (on top of the identity line, representing the perfect relationship).

```
library(ggplot2)
library(dplyr)
library(see)
data$Predicted <- predicted$Predicted
data %>%
ggplot(aes(x = Petal.Length, y = Predicted)) +
geom_line(aes(x = Petal.Length, y = Petal.Length), linetype = "dashed") +
geom_point() +
ylab("Petal.Length (predicted)") +
theme_modern()
```

It seems like our model does not perform too bad. What if we added information about the `Species`

in the model?

```
model <- stan_glm(Petal.Length ~ Sepal.Length * Species, data = data)
data$Predicted_2 <- estimate_response(model)$Predicted
```

We could now plot the second observations, based on a more complex model, as a red overlay to the previous points:

```
data %>%
ggplot() +
geom_line(aes(x = Petal.Length, y = Petal.Length), linetype = "dashed") +
geom_point(aes(x = Petal.Length, y = Predicted), color = "grey") +
geom_point(aes(x = Petal.Length, y = Predicted_2), color = "red") +
ylab("Petal.Length (predicted)") +
theme_modern()
```

The new model generated much more accurate predictions (closer from the underlying regression line).

Rather than visualizing the predictions made by a model, we are often interested in visualizing the *links*. In the model above, this would be the relationship between the response and the two predictors. This can be achieved by generating the predictions on the **data grid** of the model’s data instead of the original dataset.

We will do that to visualise the relationship between the response (`Petal.Length`

) and the predictors (`Sepal.Length`

and `Species`

).

```
predicted <- estimate_response(model, data = "grid")
iris %>%
ggplot(aes(x = Sepal.Length)) +
geom_point(aes(y = Petal.Length, color = Species)) +
geom_ribbon(data = predicted, aes(ymin = CI_low, ymax = CI_high, fill = Species), alpha = 0.3) +
geom_line(data = predicted, aes(y = Predicted, color = Species), size = 1) +
theme_modern()
```

However, you might notice that the **Credible Interval (CI) bands are quite big**. This is where `estimate_link()`

is coming in. In a traditional, frequentist, regression, the predictions are deterministic: they will always fall on the regression line. However, in a Bayesian framework, they are probabilistic. Hence here, predicting the response is not the same that predicting **the link** (*i.e.*, the regression line and the uncertainty interval associated *with this line*).

In order to facilitate visualization of links, we added `estimate_link()`

as a shortcut to `estimate_response()`

with `data = "grid"`

and, for Bayesian models, `predict = "link"`

and some smoothing by default. `estimate_response()`

would be used in the context of generating actual predictions for the existing or new data, whereas `estimate_link()`

is more relevant in the context of visualization and plotting.

```
predicted <- estimate_link(model)
iris %>%
ggplot(aes(x = Sepal.Length)) +
geom_point(aes(y = Petal.Length, color = Species)) +
geom_ribbon(data = predicted, aes(ymin = CI_low, ymax = CI_high, fill = Species), alpha = 0.3) +
geom_line(data = predicted, aes(y = Predicted, color = Species), size = 1) +
theme_modern()
```

The purpose of CI bands is to provide information about the uncertainty related to the estimation. In the Bayesian framework, the credible intervals are directly related to the shape of the posterior distribution. Thus, showing different CI levels (for instance, 69%, 89% and 99%).

```
predicted <- estimate_link(model, ci = c(0.69, .89, 0.99))
iris %>%
ggplot(aes(x = Sepal.Length)) +
geom_point(aes(y = Petal.Length, color = Species)) +
geom_ribbon(data = predicted, aes(ymin = CI_low_0.99, ymax = CI_high_0.99, fill = Species), alpha = 0.2) +
geom_ribbon(data = predicted, aes(ymin = CI_low_0.89, ymax = CI_high_0.89, fill = Species), alpha = 0.3) +
geom_ribbon(data = predicted, aes(ymin = CI_low_0.69, ymax = CI_high_0.69, fill = Species), alpha = 0.3) +
geom_line(data = predicted, aes(y = Predicted, color = Species), size = 1) +
theme_modern()
```

Instead (or in addition to) representing credible/confidence intervals, the Bayesian framework also allow to represent every individual **posterior draw**. In this case, they correspond to all possible links estimated by the model. In it a nice insight into the “true” underlying probabilities in addition to summaries like the median or the CI.

```
# Keep only 100 draws (keeping all the draws might be slower)
predicted <- estimate_link(model, keep_iterations = TRUE, iterations = 100)
# Format draws for plotting
iterations <- bayestestR::reshape_iterations(predicted)
iterations$group <- paste0(iterations$iter_group, iterations$Species)
iris %>%
ggplot(aes(x = Sepal.Length)) +
geom_point(aes(y = Petal.Length, color = Species)) +
geom_line(data = iterations, aes(y = iter_value, color = Species, group = group), alpha = 0.05) +
geom_line(data = predicted, aes(y = Predicted, color = Species), size = 1) +
theme_modern()
```

Animated hypothetical outcome plots can also be easily created with `gganimate`

:

```
library(gganimate)
p <- iris %>%
ggplot(aes(x = Sepal.Length)) +
geom_point(aes(y = Petal.Length, color = Species)) +
geom_line(data = iterations, aes(y = iter_value, color = Species, group = group)) +
theme_modern() +
transition_states(iter_group, 0, 1) +
shadow_mark(past = TRUE, future = TRUE, alpha = 1 / 20, color = "grey")
gganimate::animate(p)
```