How does `predict.randomForest` estimate class probabilities?

Question

How does randomForest package estimate class probabilities when I use predict(model, data, type = "prob")?

I was using ranger for training random forests using the probability = T argument to predict probabilities. ranger says in documentation that it:

Grow a probability forest as in Malley et al. (2012).

I simulated some data and tried both packages and obtained very different results (see code below)

So I know that it uses a different technique (then ranger) to estimate probabilities. But which one?

simulate_data <- function(n){
  X <- data.frame(matrix(runif(n*10), ncol = 10))
  Y <- data.frame(Y = rbinom(n, size = 1, prob = apply(X, 1, sum) %>%
                               pnorm(mean = 5)
                             ) %>% 
                    as.factor()

  ) 
  dplyr::bind_cols(X, Y)
}

treino <- simulate_data(10000)
teste <- simulate_data(10000)

library(ranger)
modelo_ranger <- ranger(Y ~., data = treino, 
                                num.trees = 100, 
                                mtry = floor(sqrt(10)), 
                                write.forest = T, 
                                min.node.size = 100, 
                                probability = T
                                )

modelo_randomForest <- randomForest(Y ~., data = treino,
                                    ntree = 100, 
                                    mtry = floor(sqrt(10)),
                                    nodesize = 100
                                    )

pred_ranger <- predict(modelo_ranger, teste)$predictions[,1]
pred_randomForest <- predict(modelo_randomForest, teste, type = "prob")[,2]
prob_real <- apply(teste[,1:10], 1, sum) %>% pnorm(mean = 5)

data.frame(prob_real, pred_ranger, pred_randomForest) %>%
  tidyr::gather(pacote, prob, -prob_real) %>%
  ggplot(aes(x = prob, y = prob_real)) + geom_point(size = 0.1) + facet_wrap(~pacote)

Answer 1

It's just the proportion of votes of the trees in the ensemble.

library(randomForest)

rf = randomForest(Species~., data = iris, norm.votes = TRUE, proximity = TRUE)
p1 = predict(rf, iris, type = "prob")
p2 = predict(rf, iris, type = "vote", norm.votes = TRUE)

identical(p1,p2)
#[1] TRUE

---

Alternatively, if you multiply your probabilities by ntree, you get the same result, but now in counts instead of proportions.

p1 = predict(rf, iris, type = "prob")
p2 = predict(rf, iris, type = "vote", norm.votes = FALSE)

identical(500*p1,p2)
#[1] TRUE

Answer 2

The Malley (2012) is available here: http://dx.doi.org/10.3414%2FME00-01-0052. A full reference is in the references part in the ranger documentation.

In short, each tree predicts class probabilities and these probabilities are averaged for the forest prediction. For two classes, this is equivalent to a regression forest on a 0-1 coded response.

In contrast, in randomForest with type="prob" each tree predicts a class and probabilities are calculated from these classes.

In the example here I tried to use the uniform distribution instead of the normal distribution to generate the probabilities, and here the other approach seems to perform better. I wonder if these probabilities are really the truth?

By the way, the same results as in the randomForest example above can be achieved with ranger by using classification and manual probability computation (use predict.all=TRUE in prediction).

Answer 3

If you want Out-Of-Bag probability estimates, you only can do it in randomForest package in R using model$votes. The other probability estimates are not OOB.