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

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If there have been a set of survival guidelines for knowledge scientists, amongst them must be this: All the time report uncertainty estimates along with your predictions. Nonetheless, 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 totally different random weight initializations, for various numbers of epochs, or on totally different subsets of the info. However we’d nonetheless be anxious that our methodology is sort of a bit, properly … advert hoc.

On this publish, we’ll see a each sensible in addition to theoretically grounded strategy to acquiring uncertainty estimates from neural networks. First, nevertheless, let’s shortly discuss why uncertainty is that necessary – over and above its potential to save lots of a knowledge scientist’s job.

Why uncertainty?

In a society the place automated algorithms are – and might be – entrusted with increasingly more 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 may solely work if the community’s self-indicated uncertainty actually is indicative of a better likelihood 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 have been totally different relying on whether or not the reply was right or not:

Along with quantifying uncertainty, it may possibly make sense to qualify it. Within the Bayesian deep studying literature, a distinction is usually made between epistemic uncertainty and aleatoric uncertainty (Kendall and Gal 2017). Epistemic uncertainty refers to imperfections within the mannequin – within the restrict of infinite knowledge, this sort of uncertainty ought to be reducible to 0. Aleatoric uncertainty is because of knowledge sampling and measurement processes and doesn’t rely upon the scale of the dataset.

Say we practice a mannequin for object detection. With extra knowledge, the mannequin ought to grow to be extra certain about what makes a unicycle totally different from a mountainbike. Nonetheless, 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 totally different from a unicycle any extra!

What could be the implications if we might distinguish each varieties of uncertainty? If epistemic uncertainty is excessive, we will attempt to get extra coaching knowledge. The remaining aleatoric uncertainty ought to then maintain us cautioned to consider security margins in our software.

In all probability 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 without cost as we don’t simply get level estimates (the utmost aposteriori) however the full posterior distribution. Strictly talking, in Bayesian deep studying, priors ought to be put over the weights, and the posterior be decided in response to 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 manner: by coaching a community with dropout after which, utilizing dropout at take a look at time too. At take a look at time, dropout lets us extract Monte Carlo samples from the posterior, which may then be used to approximate the true posterior distribution.

That is already excellent news, however it leaves one query open: How will we select an acceptable dropout charge? The reply is: let the community be taught 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 info given.

Moreover the predictive imply of the goal variable, it may possibly moreover be made to be taught the variance. This implies we will calculate each varieties of uncertainty, epistemic and aleatoric, independently, which is beneficial within the mild of their totally 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 will implement and take a look at the meant habits on simulated knowledge. Within the implementation, there are three issues warranting our particular consideration:

  • The wrapper class used so as to add learnable-dropout habits to a Keras layer;
  • The loss perform designed to attenuate aleatoric uncertainty; and
  • The methods we will get hold of each uncertainties at take a look at 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 may 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 methods 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 = perform(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 = perform(input_shape) {
      tremendous$construct(input_shape)
      
      self$p_logit <- tremendous$add_weight(
        identify = "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 <- personal$py_wrapper$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 = perform(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 = perform(x, masks = NULL, coaching = NULL) {
      if (self$is_mc_dropout) {
        tremendous$name(self$concrete_dropout(x))
      } else {
        k_in_train_phase(
          perform()
            tremendous$name(self$concrete_dropout(x)),
          tremendous$name(x),
          coaching = coaching
        )
      }
    }
  )
)

# perform for instantiating customized wrapper
layer_concrete_dropout <- perform(object, 
                                   layer,
                                   weight_regularizer = 1e-6,
                                   dropout_regularizer = 1e-5,
                                   init_min = 0.1,
                                   init_max = 0.1,
                                   is_mc_dropout = TRUE,
                                   identify = 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,
    identify = identify,
    trainable = trainable
  ))
}

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

First, select a price 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 concerning the frequency traits of the info. Right here, we observe 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 knowledge)
n_train <- 1000
# pattern measurement (validation knowledge)
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 methods 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 knowledge right here, however this is not a necessity
input_dim <- 1
# this too might be > 1 if we wished
output_dim <- 1
hidden_dim <- 1024

enter <- layer_input(form = input_dim)

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

Now, mannequin output is attention-grabbing: Now we have 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(models = output_dim),
  weight_regularizer = wd,
  dropout_regularizer = dd
)

log_var <- output %>% layer_concrete_dropout(
  layer_dense(models = 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 be taught totally different variances for various knowledge factors. We thus hope to have the ability to account for heteroscedasticity (totally different levels of variability) within the knowledge.

Heteroscedastic loss

Accordingly, as a substitute of imply squared error we use a value perform 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 examine, this value perform incorporates two regularization phrases:

  • First, (frac{1}{2 hat{sigma}^2_i}) downweights the high-uncertainty predictions within the loss perform. 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 certain the community doesn’t merely point out excessive uncertainty all over the place.

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

heteroscedastic_loss <- perform(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 knowledge

Now we generate some take a look at knowledge and practice the mannequin.

gen_data_1d <- perform(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 knowledge – together with these uncertainty measures that is all about!

Get hold of uncertainty estimates by way of Monte Carlo sampling

As usually in a Bayesian setup, we assemble the posterior (and thus, the posterior predictive) by way of Monte Carlo sampling. Not like in conventional use of dropout, there isn’t any change in habits between coaching and take a look at 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))
}

Bear 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 median 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 provides us uncertainty estimates individually for each prediction. How do they appear?

df <- knowledge.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)

That is attention-grabbing. The coaching knowledge (in addition to the validation knowledge) have 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 uncertain about its predictions.

That is precisely the habits we wish: Threat in routinely making use of machine studying strategies arises as a result of unanticipated variations between the coaching and take a look at (actual world) distributions. If the mannequin have been to inform us “ehm, not likely seen something like that earlier than, don’t actually know what to do” that’d be an enormously beneficial 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 beneficial – we’d know we at all times need to consider a security margin. So how does it look right here?

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

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

Now let’s do this methodology on a real-world dataset.

Mixed cycle energy plant electrical vitality output estimation

This dataset is on the market from the UCI Machine Studying Repository. We explicitly selected a regression activity with steady variables completely, to make for a easy transition from the simulated knowledge.

Within the dataset suppliers’ personal phrases

The dataset incorporates 9568 knowledge 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 Strain (AP), Relative Humidity (RH) and Exhaust Vacuum (V) to foretell the online hourly electrical vitality output (EP) of the plant.

A mixed cycle energy plant (CCPP) consists of gasoline generators (GT), steam generators (ST) and warmth restoration steam turbines. In a CCPP, the electrical energy is generated by gasoline 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 practice 5 fashions: 4 single-variable regressions and one making use of all 4 predictors. It in all probability 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 info

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 number 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 <- perform(input_dim, hidden_dim) {
  
  enter <- layer_input(form = input_dim)
  output <-
    enter %>% layer_concrete_dropout(
      layer = layer_dense(models = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    ) %>% layer_concrete_dropout(
      layer = layer_dense(models = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    ) %>% layer_concrete_dropout(
      layer = layer_dense(models = hidden_dim, activation = "relu"),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  imply <-
    output %>% layer_concrete_dropout(
      layer = layer_dense(models = output_dim),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  log_var <-
    output %>% layer_concrete_dropout(
      layer_dense(models = output_dim),
      weight_regularizer = wd,
      dropout_regularizer = dd
    )
  
  output <- layer_concatenate(record(imply, log_var))
  
  mannequin <- keras_model(enter, output)
  
  heteroscedastic_loss <- perform(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 practice 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 related 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 <- knowledge.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)
)

Outcome

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

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

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

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

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, acquired so much decrease. General, predictive uncertainty captures the vary of floor fact values fairly properly.

Conclusion

We’ve launched a way to acquire theoretically grounded uncertainty estimates from neural networks. We discover the strategy intuitively engaging for a number of causes: For one, the separation of several types of uncertainty is convincing and virtually related. Second, uncertainty depends upon the quantity of knowledge seen within the respective ranges. That is particularly related when pondering of variations between coaching and test-time distributions. Third, the thought of getting the community “grow to be conscious of its personal uncertainty” is seductive.

In follow although, there are open questions as to methods to apply the strategy. From our real-world take a look at above, we instantly ask: Why is the mannequin so assured when the bottom fact knowledge has excessive variance? And, pondering experimentally: How would that modify with totally different knowledge sizes (rows), dimensionality (columns), and hyperparameter settings (together with neural community hyperparameters like capability, variety of epochs educated, and activation capabilities, but additionally the Gaussian course of prior length-scale (tau))?

For sensible use, extra experimentation with totally different datasets and hyperparameter settings is definitely warranted. One other course to observe up is software to duties in picture recognition, resembling semantic segmentation. Right here we’d be fascinated with not simply quantifying, but additionally localizing uncertainty, to see which visible elements of a scene (occlusion, illumination, unusual shapes) make objects onerous to establish.

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 Laptop Imaginative and prescient?” In Advances in Neural Data Processing Techniques 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 Data from Deep Neural Networks for Illness Detection.” bioRxiv. https://doi.org/10.1101/084210.

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