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You positive? A Bayesian strategy to acquiring uncertainty estimates from neural networks


If there have been a set of survival guidelines for information scientists, amongst them must be this: At all times report uncertainty estimates along with your predictions. Nevertheless, right here we’re, working with neural networks, and in contrast to lm, a Keras mannequin doesn’t conveniently output one thing like a normal error for the weights. We’d attempt to think about rolling your personal uncertainty measure – for instance, averaging predictions from networks educated from completely different random weight initializations, for various numbers of epochs, or on completely different subsets of the information. However we’d nonetheless be anxious that our technique is kind of a bit, properly … advert hoc.

On this submit, we’ll see a each sensible in addition to theoretically grounded strategy to acquiring uncertainty estimates from neural networks. First, nonetheless, let’s shortly speak about why uncertainty is that essential – over and above its potential to avoid wasting a knowledge scientist’s job.

Why uncertainty?

In a society the place automated algorithms are – and will likely be – entrusted with increasingly life-critical duties, one reply instantly jumps to thoughts: If the algorithm accurately quantifies its uncertainty, we might have human specialists examine the extra unsure predictions and probably revise them.

This can solely work if the community’s self-indicated uncertainty actually is indicative of a better chance of misclassification. Leibig et al.(Leibig et al. 2017) used a predecessor of the strategy described beneath to evaluate neural community uncertainty in detecting diabetic retinopathy. They discovered that certainly, the distributions of uncertainty had been completely different relying on whether or not the reply was appropriate or not:

Along with quantifying uncertainty, it may well make sense to qualify it. Within the Bayesian deep studying literature, a distinction is often made between epistemic uncertainty and aleatoric uncertainty (Kendall and Gal 2017). Epistemic uncertainty refers to imperfections within the mannequin – within the restrict of infinite information, this type of uncertainty needs to be reducible to 0. Aleatoric uncertainty is because of information sampling and measurement processes and doesn’t depend upon the dimensions of the dataset.

Say we prepare a mannequin for object detection. With extra information, the mannequin ought to change into extra positive about what makes a unicycle completely different from a mountainbike. Nevertheless, let’s assume all that’s seen of the mountainbike is the entrance wheel, the fork and the pinnacle tube. Then it doesn’t look so completely different from a unicycle any extra!

What could be the implications if we might distinguish each forms of uncertainty? If epistemic uncertainty is excessive, we are able to attempt to get extra coaching information. The remaining aleatoric uncertainty ought to then preserve us cautioned to think about security margins in our software.

Most likely no additional justifications are required of why we’d wish to assess mannequin uncertainty – however how can we do that?

Uncertainty estimates via Bayesian deep studying

In a Bayesian world, in precept, uncertainty is free of charge as we don’t simply get level estimates (the utmost aposteriori) however the full posterior distribution. Strictly talking, in Bayesian deep studying, priors needs to be put over the weights, and the posterior be decided in line with Bayes’ rule. To the deep studying practitioner, this sounds fairly arduous – and the way do you do it utilizing Keras?

In 2016 although, Gal and Ghahramani (Yarin Gal and Ghahramani 2016) confirmed that when viewing a neural community as an approximation to a Gaussian course of, uncertainty estimates may be obtained in a theoretically grounded but very sensible method: by coaching a community with dropout after which, utilizing dropout at check time too. At check time, dropout lets us extract Monte Carlo samples from the posterior, which might then be used to approximate the true posterior distribution.

That is already excellent news, nevertheless it leaves one query open: How can we select an applicable dropout charge? The reply is: let the community study it.

Studying dropout and uncertainty

In a number of 2017 papers (Y. Gal, Hron, and Kendall 2017),(Kendall and Gal 2017), Gal and his coworkers demonstrated how a community may be educated to dynamically adapt the dropout charge so it’s sufficient for the quantity and traits of the information given.

Moreover the predictive imply of the goal variable, it may well moreover be made to study the variance. This implies we are able to calculate each forms of uncertainty, epistemic and aleatoric, independently, which is helpful within the gentle of their completely different implications. We then add them as much as get hold of the general predictive uncertainty.

Let’s make this concrete and see how we are able to implement and check the meant conduct on simulated information. Within the implementation, there are three issues warranting our particular consideration:

  • The wrapper class used so as to add learnable-dropout conduct to a Keras layer;
  • The loss operate designed to reduce aleatoric uncertainty; and
  • The methods we are able to get hold of each uncertainties at check time.

Let’s begin with the wrapper.

A wrapper for studying dropout

On this instance, we’ll prohibit ourselves to studying dropout for dense layers. Technically, we’ll add a weight and a loss to each dense layer we wish to use dropout with. This implies we’ll create a customized wrapper class that has entry to the underlying layer and might modify it.

The logic applied within the wrapper is derived mathematically within the Concrete Dropout paper (Y. Gal, Hron, and Kendall 2017). The beneath code is a port to R of the Python Keras model discovered within the paper’s companion github repo.

So first, right here is the wrapper class – we’ll see learn how to use it in only a second:

library(keras)

# R6 wrapper class, a subclass of KerasWrapper
ConcreteDropout <- R6::R6Class("ConcreteDropout",
  
  inherit = KerasWrapper,
  
  public = record(
    weight_regularizer = NULL,
    dropout_regularizer = NULL,
    init_min = NULL,
    init_max = NULL,
    is_mc_dropout = NULL,
    supports_masking = TRUE,
    p_logit = NULL,
    p = NULL,
    
    initialize = operate(weight_regularizer,
                          dropout_regularizer,
                          init_min,
                          init_max,
                          is_mc_dropout) {
      self$weight_regularizer <- weight_regularizer
      self$dropout_regularizer <- dropout_regularizer
      self$is_mc_dropout <- is_mc_dropout
      self$init_min <- k_log(init_min) - k_log(1 - init_min)
      self$init_max <- k_log(init_max) - k_log(1 - init_max)
    },
    
    construct = operate(input_shape) {
      tremendous$construct(input_shape)
      
      self$p_logit <- tremendous$add_weight(
        title = "p_logit",
        form = form(1),
        initializer = initializer_random_uniform(self$init_min, self$init_max),
        trainable = TRUE
      )

      self$p <- k_sigmoid(self$p_logit)

      input_dim <- input_shape[[2]]

      weight <- non-public$py_wrapper$layer$kernel
      
      kernel_regularizer <- self$weight_regularizer * 
                            k_sum(k_square(weight)) / 
                            (1 - self$p)
      
      dropout_regularizer <- self$p * k_log(self$p)
      dropout_regularizer <- dropout_regularizer +  
                             (1 - self$p) * k_log(1 - self$p)
      dropout_regularizer <- dropout_regularizer * 
                             self$dropout_regularizer * 
                             k_cast(input_dim, k_floatx())

      regularizer <- k_sum(kernel_regularizer + dropout_regularizer)
      tremendous$add_loss(regularizer)
    },
    
    concrete_dropout = operate(x) {
      eps <- k_cast_to_floatx(k_epsilon())
      temp <- 0.1
      
      unif_noise <- k_random_uniform(form = k_shape(x))
      
      drop_prob <- k_log(self$p + eps) - 
                   k_log(1 - self$p + eps) + 
                   k_log(unif_noise + eps) - 
                   k_log(1 - unif_noise + eps)
      drop_prob <- k_sigmoid(drop_prob / temp)
      
      random_tensor <- 1 - drop_prob
      
      retain_prob <- 1 - self$p
      x <- x * random_tensor
      x <- x / retain_prob
      x
    },

    name = operate(x, masks = NULL, coaching = NULL) {
      if (self$is_mc_dropout) {
        tremendous$name(self$concrete_dropout(x))
      } else {
        k_in_train_phase(
          operate()
            tremendous$name(self$concrete_dropout(x)),
          tremendous$name(x),
          coaching = coaching
        )
      }
    }
  )
)

# operate for instantiating customized wrapper
layer_concrete_dropout <- operate(object, 
                                   layer,
                                   weight_regularizer = 1e-6,
                                   dropout_regularizer = 1e-5,
                                   init_min = 0.1,
                                   init_max = 0.1,
                                   is_mc_dropout = TRUE,
                                   title = NULL,
                                   trainable = TRUE) {
  create_wrapper(ConcreteDropout, object, record(
    layer = layer,
    weight_regularizer = weight_regularizer,
    dropout_regularizer = dropout_regularizer,
    init_min = init_min,
    init_max = init_max,
    is_mc_dropout = is_mc_dropout,
    title = title,
    trainable = trainable
  ))
}

The wrapper instantiator has default arguments, however two of them needs to be tailored to the information: weight_regularizer and dropout_regularizer. Following the authors’ suggestions, they need to be set as follows.

First, select a worth for hyperparameter (l). On this view of a neural community as an approximation to a Gaussian course of, (l) is the prior length-scale, our a priori assumption in regards to the frequency traits of the information. Right here, we comply with Gal’s demo in setting l := 1e-4. Then the preliminary values for weight_regularizer and dropout_regularizer are derived from the length-scale and the pattern measurement.

# pattern measurement (coaching information)
n_train <- 1000
# pattern measurement (validation information)
n_val <- 1000
# prior length-scale
l <- 1e-4
# preliminary worth for weight regularizer 
wd <- l^2/n_train
# preliminary worth for dropout regularizer
dd <- 2/n_train

Now let’s see learn how to use the wrapper in a mannequin.

Dropout mannequin

In our demonstration, we’ll have a mannequin with three hidden dense layers, every of which could have its dropout charge calculated by a devoted wrapper.

# we use one-dimensional enter information right here, however this is not a necessity
input_dim <- 1
# this too could possibly be > 1 if we needed
output_dim <- 1
hidden_dim <- 1024

enter <- layer_input(form = input_dim)

output <- enter %>% layer_concrete_dropout(
  layer = layer_dense(items = hidden_dim, activation = "relu"),
  weight_regularizer = wd,
  dropout_regularizer = dd
  ) %>% layer_concrete_dropout(
  layer = layer_dense(items = hidden_dim, activation = "relu"),
  weight_regularizer = wd,
  dropout_regularizer = dd
  ) %>% layer_concrete_dropout(
  layer = layer_dense(items = hidden_dim, activation = "relu"),
  weight_regularizer = wd,
  dropout_regularizer = dd
)

Now, mannequin output is attention-grabbing: We’ve got the mannequin yielding not simply the predictive (conditional) imply, but additionally the predictive variance ((tau^{-1}) in Gaussian course of parlance):

imply <- output %>% layer_concrete_dropout(
  layer = layer_dense(items = output_dim),
  weight_regularizer = wd,
  dropout_regularizer = dd
)

log_var <- output %>% layer_concrete_dropout(
  layer_dense(items = output_dim),
  weight_regularizer = wd,
  dropout_regularizer = dd
)

output <- layer_concatenate(record(imply, log_var))

mannequin <- keras_model(enter, output)

The numerous factor right here is that we study completely different variances for various information factors. We thus hope to have the ability to account for heteroscedasticity (completely different levels of variability) within the information.

Heteroscedastic loss

Accordingly, as a substitute of imply squared error we use a price operate that doesn’t deal with all estimates alike(Kendall and Gal 2017):

[frac{1}{N} sum_i{frac{1}{2 hat{sigma}^2_i} (mathbf{y}_i – mathbf{hat{y}}_i)^2 + frac{1}{2} log hat{sigma}^2_i}]

Along with the compulsory goal vs. prediction test, this value operate incorporates two regularization phrases:

  • First, (frac{1}{2 hat{sigma}^2_i}) downweights the high-uncertainty predictions within the loss operate. Put plainly: The mannequin is inspired to point excessive uncertainty when its predictions are false.
  • Second, (frac{1}{2} log hat{sigma}^2_i) makes positive the community doesn’t merely point out excessive uncertainty all over the place.

This logic maps on to the code (besides that as standard, we’re calculating with the log of the variance, for causes of numerical stability):

heteroscedastic_loss <- operate(y_true, y_pred) {
    imply <- y_pred[, 1:output_dim]
    log_var <- y_pred[, (output_dim + 1):(output_dim * 2)]
    precision <- k_exp(-log_var)
    k_sum(precision * (y_true - imply) ^ 2 + log_var, axis = 2)
  }

Coaching on simulated information

Now we generate some check information and prepare the mannequin.

gen_data_1d <- operate(n) {
  sigma <- 1
  X <- matrix(rnorm(n))
  w <- 2
  b <- 8
  Y <- matrix(X %*% w + b + sigma * rnorm(n))
  record(X, Y)
}

c(X, Y) %<-% gen_data_1d(n_train + n_val)

c(X_train, Y_train) %<-% record(X[1:n_train], Y[1:n_train])
c(X_val, Y_val) %<-% record(X[(n_train + 1):(n_train + n_val)], 
                          Y[(n_train + 1):(n_train + n_val)])

mannequin %>% compile(
  optimizer = "adam",
  loss = heteroscedastic_loss,
  metrics = c(custom_metric("heteroscedastic_loss", heteroscedastic_loss))
)

historical past <- mannequin %>% match(
  X_train,
  Y_train,
  epochs = 30,
  batch_size = 10
)

With coaching completed, we flip to the validation set to acquire estimates on unseen information – together with these uncertainty measures that is all about!

Get hold of uncertainty estimates through Monte Carlo sampling

As usually in a Bayesian setup, we assemble the posterior (and thus, the posterior predictive) through Monte Carlo sampling. Not like in conventional use of dropout, there is no such thing as a change in conduct between coaching and check phases: Dropout stays “on.”

So now we get an ensemble of mannequin predictions on the validation set:

num_MC_samples <- 20

MC_samples <- array(0, dim = c(num_MC_samples, n_val, 2 * output_dim))
for (okay in 1:num_MC_samples) {
  MC_samples[k, , ] <- (mannequin %>% predict(X_val))
}

Keep in mind, our mannequin predicts the imply in addition to the variance. We’ll use the previous for calculating epistemic uncertainty, whereas aleatoric uncertainty is obtained from the latter.

First, we decide the predictive imply as a mean of the MC samples’ imply output:

# the means are within the first output column
means <- MC_samples[, , 1:output_dim]  
# common over the MC samples
predictive_mean <- apply(means, 2, imply) 

To calculate epistemic uncertainty, we once more use the imply output, however this time we’re within the variance of the MC samples:

epistemic_uncertainty <- apply(means, 2, var) 

Then aleatoric uncertainty is the typical over the MC samples of the variance output..

logvar <- MC_samples[, , (output_dim + 1):(output_dim * 2)]
aleatoric_uncertainty <- exp(colMeans(logvar))

Observe how this process offers us uncertainty estimates individually for each prediction. How do they give the impression of being?

df <- information.body(
  x = X_val,
  y_pred = predictive_mean,
  e_u_lower = predictive_mean - sqrt(epistemic_uncertainty),
  e_u_upper = predictive_mean + sqrt(epistemic_uncertainty),
  a_u_lower = predictive_mean - sqrt(aleatoric_uncertainty),
  a_u_upper = predictive_mean + sqrt(aleatoric_uncertainty),
  u_overall_lower = predictive_mean - 
                    sqrt(epistemic_uncertainty) - 
                    sqrt(aleatoric_uncertainty),
  u_overall_upper = predictive_mean + 
                    sqrt(epistemic_uncertainty) + 
                    sqrt(aleatoric_uncertainty)
)

Right here, first, is epistemic uncertainty, with shaded bands indicating one normal deviation above resp. beneath the anticipated imply:

ggplot(df, aes(x, y_pred)) + 
  geom_point() + 
  geom_ribbon(aes(ymin = e_u_lower, ymax = e_u_upper), alpha = 0.3)
Epistemic uncertainty on the validation set, train size = 1000.

That is attention-grabbing. The coaching information (in addition to the validation information) had been generated from a typical regular distribution, so the mannequin has encountered many extra examples near the imply than outdoors two, and even three, normal deviations. So it accurately tells us that in these extra unique areas, it feels fairly not sure about its predictions.

That is precisely the conduct we would like: Threat in robotically making use of machine studying strategies arises as a result of unanticipated variations between the coaching and check (actual world) distributions. If the mannequin had been to inform us “ehm, probably not seen something like that earlier than, don’t actually know what to do” that’d be an enormously worthwhile end result.

So whereas epistemic uncertainty has the algorithm reflecting on its mannequin of the world – probably admitting its shortcomings – aleatoric uncertainty, by definition, is irreducible. After all, that doesn’t make it any much less worthwhile – we’d know we all the time should think about a security margin. So how does it look right here?

Aleatoric uncertainty on the validation set, train size = 1000.

Certainly, the extent of uncertainty doesn’t depend upon the quantity of information seen at coaching time.

Lastly, we add up each sorts to acquire the general uncertainty when making predictions.

Overall predictive uncertainty on the validation set, train size = 1000.

Now let’s do that technique on a real-world dataset.

Mixed cycle energy plant electrical vitality output estimation

This dataset is obtainable from the UCI Machine Studying Repository. We explicitly selected a regression activity with steady variables completely, to make for a clean transition from the simulated information.

Within the dataset suppliers’ personal phrases

The dataset incorporates 9568 information factors collected from a Mixed Cycle Energy Plant over 6 years (2006-2011), when the facility plant was set to work with full load. Options include hourly common ambient variables Temperature (T), Ambient Stress (AP), Relative Humidity (RH) and Exhaust Vacuum (V) to foretell the web hourly electrical vitality output (EP) of the plant.

A mixed cycle energy plant (CCPP) consists of fuel generators (GT), steam generators (ST) and warmth restoration steam turbines. In a CCPP, the electrical energy is generated by fuel and steam generators, that are mixed in a single cycle, and is transferred from one turbine to a different. Whereas the Vacuum is collected from and has impact on the Steam Turbine, the opposite three of the ambient variables impact the GT efficiency.

We thus have 4 predictors and one goal variable. We’ll prepare 5 fashions: 4 single-variable regressions and one making use of all 4 predictors. It most likely goes with out saying that our aim right here is to examine uncertainty info, to not fine-tune the mannequin.

Setup

Let’s shortly examine these 5 variables. Right here PE is vitality output, the goal variable.

We scale and divide up the information

df_scaled <- scale(df)

X <- df_scaled[, 1:4]
train_samples <- pattern(1:nrow(df_scaled), 0.8 * nrow(X))
X_train <- X[train_samples,]
X_val <- X[-train_samples,]

y <- df_scaled[, 5] %>% as.matrix()
y_train <- y[train_samples,]
y_val <- y[-train_samples,]

and prepare for coaching a couple of fashions.

n <- nrow(X_train)
n_epochs <- 100
batch_size <- 100
output_dim <- 1
num_MC_samples <- 20
l <- 1e-4
wd <- l^2/n
dd <- 2/n

get_model <- operate(input_dim, hidden_dim) {
  
  enter <- layer_input(form = input_dim)
  output <-
    enter %>% layer_concrete_dropout(
      layer = layer_dense(items = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    ) %>% layer_concrete_dropout(
      layer = layer_dense(items = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    ) %>% layer_concrete_dropout(
      layer = layer_dense(items = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  imply <-
    output %>% layer_concrete_dropout(
      layer = layer_dense(items = output_dim),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  log_var <-
    output %>% layer_concrete_dropout(
      layer_dense(items = output_dim),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  output <- layer_concatenate(record(imply, log_var))
  
  mannequin <- keras_model(enter, output)
  
  heteroscedastic_loss <- operate(y_true, y_pred) {
    imply <- y_pred[, 1:output_dim]
    log_var <- y_pred[, (output_dim + 1):(output_dim * 2)]
    precision <- k_exp(-log_var)
    k_sum(precision * (y_true - imply) ^ 2 + log_var, axis = 2)
  }
  
  mannequin %>% compile(optimizer = "adam",
                    loss = heteroscedastic_loss,
                    metrics = c("mse"))
  mannequin
}

We’ll prepare every of the 5 fashions with a hidden_dim of 64. We then get hold of 20 Monte Carlo pattern from the posterior predictive distribution and calculate the uncertainties as earlier than.

Right here we present the code for the primary predictor, “AT.” It’s comparable for all different instances.

mannequin <- get_model(1, 64)
hist <- mannequin %>% match(
  X_train[ ,1],
  y_train,
  validation_data = record(X_val[ , 1], y_val),
  epochs = n_epochs,
  batch_size = batch_size
)

MC_samples <- array(0, dim = c(num_MC_samples, nrow(X_val), 2 * output_dim))
for (okay in 1:num_MC_samples) {
  MC_samples[k, ,] <- (mannequin %>% predict(X_val[ ,1]))
}

means <- MC_samples[, , 1:output_dim]  
predictive_mean <- apply(means, 2, imply) 
epistemic_uncertainty <- apply(means, 2, var) 
logvar <- MC_samples[, , (output_dim + 1):(output_dim * 2)]
aleatoric_uncertainty <- exp(colMeans(logvar))

preds <- information.body(
  x1 = X_val[, 1],
  y_true = y_val,
  y_pred = predictive_mean,
  e_u_lower = predictive_mean - sqrt(epistemic_uncertainty),
  e_u_upper = predictive_mean + sqrt(epistemic_uncertainty),
  a_u_lower = predictive_mean - sqrt(aleatoric_uncertainty),
  a_u_upper = predictive_mean + sqrt(aleatoric_uncertainty),
  u_overall_lower = predictive_mean - 
                    sqrt(epistemic_uncertainty) - 
                    sqrt(aleatoric_uncertainty),
  u_overall_upper = predictive_mean + 
                    sqrt(epistemic_uncertainty) + 
                    sqrt(aleatoric_uncertainty)
)

Consequence

Now let’s see the uncertainty estimates for all 5 fashions!

First, the single-predictor setup. Floor reality values are displayed in cyan, posterior predictive estimates are black, and the gray bands prolong up resp. down by the sq. root of the calculated uncertainties.

We’re beginning with ambient temperature, a low-variance predictor. We’re shocked how assured the mannequin is that it’s gotten the method logic appropriate, however excessive aleatoric uncertainty makes up for this (roughly).

Uncertainties on the validation set using ambient temperature as a single predictor.

Now wanting on the different predictors, the place variance is far greater within the floor reality, it does get a bit troublesome to really feel comfy with the mannequin’s confidence. Aleatoric uncertainty is excessive, however not excessive sufficient to seize the true variability within the information. And we certaintly would hope for greater epistemic uncertainty, particularly in locations the place the mannequin introduces arbitrary-looking deviations from linearity.

Uncertainties on the validation set using exhaust vacuum as a single predictor.
Uncertainties on the validation set using ambient pressure as a single predictor.
Uncertainties on the validation set using relative humidity as a single predictor.

Now let’s see uncertainty output after we use all 4 predictors. We see that now, the Monte Carlo estimates fluctuate much more, and accordingly, epistemic uncertainty is so much greater. Aleatoric uncertainty, however, received so much decrease. General, predictive uncertainty captures the vary of floor reality values fairly properly.

Uncertainties on the validation set using all 4 predictors.

Conclusion

We’ve launched a way to acquire theoretically grounded uncertainty estimates from neural networks. We discover the strategy intuitively enticing for a number of causes: For one, the separation of various kinds of uncertainty is convincing and virtually related. Second, uncertainty will depend on the quantity of information seen within the respective ranges. That is particularly related when pondering of variations between coaching and test-time distributions. Third, the concept of getting the community “change into conscious of its personal uncertainty” is seductive.

In follow although, there are open questions as to learn how to apply the strategy. From our real-world check above, we instantly ask: Why is the mannequin so assured when the bottom reality information has excessive variance? And, pondering experimentally: How would that change with completely different information sizes (rows), dimensionality (columns), and hyperparameter settings (together with neural community hyperparameters like capability, variety of epochs educated, and activation features, but additionally the Gaussian course of prior length-scale (tau))?

For sensible use, extra experimentation with completely different datasets and hyperparameter settings is definitely warranted. One other route to comply with up is software to duties in picture recognition, equivalent to semantic segmentation. Right here we’d be curious about not simply quantifying, but additionally localizing uncertainty, to see which visible points of a scene (occlusion, illumination, unusual shapes) make objects arduous to determine.

Gal, Yarin, and Zoubin Ghahramani. 2016. “Dropout as a Bayesian Approximation: Representing Mannequin Uncertainty in Deep Studying.” In Proceedings of the 33nd Worldwide Convention on Machine Studying, ICML 2016, New York Metropolis, NY, USA, June 19-24, 2016, 1050–59. http://jmlr.org/proceedings/papers/v48/gal16.html.
Gal, Y., J. Hron, and A. Kendall. 2017. “Concrete Dropout.” ArXiv e-Prints, Could. https://arxiv.org/abs/1705.07832.
Kendall, Alex, and Yarin Gal. 2017. “What Uncertainties Do We Want in Bayesian Deep Studying for Pc Imaginative and prescient?” In Advances in Neural Info Processing Programs 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5574–84. Curran Associates, Inc. http://papers.nips.cc/paper/7141-what-uncertainties-do-we-need-in-bayesian-deep-learning-for-computer-vision.pdf.
Leibig, Christian, Vaneeda Allken, Murat Seckin Ayhan, Philipp Berens, and Siegfried Wahl. 2017. “Leveraging Uncertainty Info from Deep Neural Networks for Illness Detection.” bioRxiv. https://doi.org/10.1101/084210.

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