Training models from data

Here we provide a non-exhaustive review of methods to calibrate probabilistic and non- probabilistic generative uncertainty models, in order to set the context for the remainder of the thesis.

Creating parametric Bayesian probabilistic models

Consider the probability distribution pθ(x) = p(x(i)|θ) with vector of parameters θ, which we wish to identify based on a set of n training samples, _Xtrain={x(1), . . . , x(n)}, drawn from the random variable specified by pθ(x). A distribution over the parameters θ, given the data _Xtrain can be obtained by applying Bayes’ law:

P (θ_|Xtrain) =

P (_Xtrain|θ)p(θ) P (_Xtrain)

, (2.21)

where p(θ) represents a prior distribution on θ, P (_Xtrain) = R P (Xtrain|θ)dθ acts as a normalising constant, and the data likelihood can be written as P (Xtrain|θ) =Qip(x(i)|θ) by assuming independence of training samples. This approach, known as Bayesian Hierarchical Modelling [74], has desirable properties. For example, the epistemic uncertainty on θ will decrease as more data becomes available which will be observed as a ‘concentration’ of the posterior distribution for θ around one point.

Although simple analytical distributions are often used for the likelihood P (Xtrain|θ) (e.g. a Gaussian distribution with mean θ1 and scale θ2). One can also extend the framework to consider more complex likelihood functions. For example, often the likelihood

22 Jonathan Cyrus Sadeghi

is P (Xtrain|θ) = R P (Xtrain|y)δ(f(θ) − y)dy, where f(θ) is an arbitrarily expensive and complex function, for which we may not know the gradient, and p(x(i)_{|θ) is a simple} probability density, for example a normal distribution, and δ(x) is the Dirac delta function. This setting is referred to as an ‘inverse problem’ [169].

The probability distributions over θ represent epistemic uncertainty in θ, whilst the data likelihood, p(x(i)_{|θ), represents the natural stochasticity (aleatory uncertainty) of the data} generating mechanism. Note that the prior distribution, p(θ), should be chosen to represent our prior knowledge of the parameter θ, and in the case of no knowledge, should be set to an appropriate uninformative distribution. The distribution used for the uninformative prior should be chosen based on physical considerations regarding the parameter of interest, but is often somewhat arbitrarily assumed to be uniform [88].

The prior distribution is not the only place where prior knowledge enters into the probabilistic model; the model specification, i.e. the data likelihood, represents another form of prior knowledge which must be carefully considered with this approach [88]. It is particularly important to decide which parameters in the likelihood function should be modelled as uncertain, e.g. if the likelihood is assumed to be a Gaussian density, will a value be assumed for the standard deviation of the distribution, or will this be an element of θ, and hence an uncertain parameter?

We can derive point estimates for θ from the Bayesian approach [67]. The maximum a posteriori estimator for θ is obtained by evaluating θMAP = maxθP (θ|Xtrain), where P (θ|Xtrain)∝ P (Xtrain|θ)P (θ). The maximum likelihood estimator for θ is obtained by eval- uating θML= maxθP (Xtrain|θ). Note that the maximum likelihood estimator is equivalent to the maximum a posteriori estimator when a uniform prior distribution is used. These estimators can be evaluated using any optimisation method. Stochastic Gradent Descent, a widely used optimisation method, will be discussed in Chapter 3 since it is most often applied to regression models.

We do not necessarily have to disregard uncertainty in θ when using the maximum a posteriori approach, since the covariance of the distribution can be estimated by inverting the Hessian (matrix of second derivatives with respect to parameters θ) of P (θ_|Xtrain). This is known as the Laplace approximation. This estimate is exact in some well known cases, e.g. the case of a Gaussian likelihood and prior, where the optimisation loss (the logarithm of the posterior) becomes the mean squared error [169].

In many cases the full posterior for θ can be calculated analytically, for example where a conjugate prior is used so that the posterior distribution has the same functional form as

Chapter 2. Models of Uncertainty 23

the prior distribution. If this is not the case, and one wishes to compute the full posterior distribution, then a Markov Chain Monte Carlo (MCMC) method is often used to obtain samples from the posterior distribution, or other approximate numerical techniques are used.

An MCMC algorithm constructs a Markov chain with the desired distribution as its equilibrium distribution, so that if the Markov chain is simulated for a sufficient time then the samples drawn are from the posterior distribution [74]. MCMC methods typically do not require the gradient of the posterior to be known, and are hence applicable to a wide class of problems. Unfortunately, MCMC simulation can be computationally infeasible when θ has high dimensionality, or when the training dataset is large. Recently, efficient sampling based algorithms have been proposed to combat this problem [80].

As an alternative to MCMC based methods, Variational Inference can be used to find the closest match between an approximating parametric ‘proposal’ distribution and the true posterior distribution. This method typically requires the gradient of the likelihood function to be known, but scales very well to high dimensionality problems [21].

Approximate Bayesian Computation is an efficient computational method which can be used to sample from an approximation of the posterior distribution in the case that the likelihood is too expensive to compute [48]. Sadeghi et al. [154] demonstrate a similar method, where the true likelihood probability density function is replaced by an interval with associated probability, and show that bounds on the likelihood function can still be obtained in this case.

Frequentist confidence intervals

In this thesis, traditional frequentist statistics are not used, except for in the validation of some Interval Predictor Models, but for the interested reader we briefly describe here how a frequentist confidence interval can be obtained for θ.

In frequentist statistics, one aims to identify a region of parameter space which would contain the true value of the parameter with a specified frequency if the experiment was repeated, i.e. we aim to find the confidence interval Θ = [θ, θ], where P (θ_{∈ Θ) = 1 − α,} and α is an arbitrarily small probability.

24 Jonathan Cyrus Sadeghi

Non-parametric prediction intervals

The maximum and minimum of a dataset (i.e. [minix(i), maxix(i)] when x is one dimen- sional) can be used to produce a prediction interval with coverage probability n−1_n+1, i.e. the probability that x(n+1) will fall inside the prediction interval [183]. A tighter prediction interval, with a lower coverage probability of n+1−2j_n+1 can be obtained by using the j-th smallest and largest values in the dataset.

Creating parametric imprecise probability models

The application of Bayes’ law in Eqn. 2.21 assumes that the data _Xtrain consists of real, ‘crisp’, values. However, we can also apply Bayes’ law to imprecise, interval data. For

example, consider the set of training data _Xtrain = {[x(1), x(1)], . . . , [x(n), x(n)]}. If an analytic equation is available for the posterior parameters then in many cases it is possible to obtain bounds on the posterior parameters given interval data. For example, if a Gaussian density is used for the likelihood and prior, then one may obtain bounds on the posterior normal distribution parameters analytically [120].

The standard Bayesian paradigm can also be made robust by considering a set of prior distributions. This is known as Robust Bayes [17]. Again, bounds on the posterior parameters are available analytically in many cases, e.g. the Imprecise Dirichlet model.

Probability boxes can also be obtained by creating so-called confidence structures, which are encoded as probability boxes. Confidence boxes encode confidence intervals at all confidence levels. The binomial confidence bounds, which bound the success probability of a binomial random variable, are a particularly useful example which can be found by inverting the CDF of a binomial random variable [62].

Creating non-parametric imprecise probability models

Several methods exist to obtain non-parametric CDFs from data. The CDF can be estimated from n training samples,_Xtrain={x(1), . . . , x(n)}, using the empirical cumulative distribution function (eCDF), which is given by

Sn(x) = 1 n n X i Ix≥x(i)(x), (2.22)

Chapter 2. Models of Uncertainty 25

where I is the indicator function, which is equal to 1 if the subscript statement is satisfied, and is otherwise equal to zero [101]. The eCDF is effectively the random variable which is formed by assigning probability density equally at the point value of each sample, and hence when plotted the eCDF looks like a staircase function. The eCDF can be generalised to the case of imprecise sampled data, by considering a separate eCDF for the upper and lower bounds of the samples. These upper and lower bounds represent the envelope of a probability box, and hence an empirical probability box is obtained [60].

So-called concentration inequalities can be used to obtain bounds on the CDF of a random variable with a certain confidence. A probability box can be obtained for the random variable by choosing a cutoff confidence, such that the CDFs at that confidence will form the envelope of the probability box [60].

The Kolmogorov-Smirnov statistic can be used to measure the confidence that the true CDF of a random variable differs by more than a certain probability from the eCDF obtained from sampled data (i.e. the vertical distance between the CDFs is compared) [126]. The Kolmogorov-Smirnov statistic is given by

D = sup x |Sn

(x)_{− F (x)|,} (2.23)

where F (x) is the true CDF and the values for D can be obtained from Kolmogoroff [101]. Now a set of CDFs can be found with associated confidence. The Kolmogorov-Smirnov statistic can be applied to eCDF bounds obtained by considering imprecise data [60].

Chebyshev’s inequality bounds the probability density of a random variable which can fall more than a certain number of standard deviations from the mean [149]. Therefore knowledge of the mean and standard deviation of a random variable imposes bounds on its CDF. Chebyshev’s inequality is given by

P ((X− µ) ≥ kσ) ≤ 1

k2, (2.24)

where k > 1 is a real number, µ is the mean of a random variable, and σ is the standard deviation of the random variable.