# Uncertainty in Engineering Introduction to Methods and Applications

Abstract We present basic concepts of Bayesian statistical inference. We briefly introduce the Bayesian paradigm. We present the conjugate priors; a computational convenient way to quantify prior information for tractable Bayesian statistical analysis. We present tools for parametric and predictive inference, and particularly the design of point estimators, credible sets, and hypothesis tests. These concepts are presented in running examples. Supplementary material is available from GitHub.

1.1 Introduction
Statistics mainly aim at addressing two major things. First, we wish to learn or draw conclusions about an unknown quantity, θ ∈  called ‘the parameter’, which cannot be directly measured or observed, by measuring or observing a sequence of other quantities called ‘observations (or data, or samples)’ x1:n := (x1,..., xn) ∈ Xm
whose generating mechanism is (or can be considered as) stochastically dependent on the quantity of interest θ though a probabilistic model x1:n ∼ f (·|θ ). This is an inverse problem since we wish to study the cause θ by knowing its effect x1:n. We will refer to this as parametric inference. Second, we wish to learn the possible values of a future sequence of observations y1:m ∈ Xm given x1:n. This is a forward problem,and we will call it predictive inference. Here, we present how both inferences can be
Consider a sequence of observables x1:n := (x1,..., xn) generated from a sampling distribution f (·|θ ) labeled by the unknown parameter θ ∈ . The statistical model m consists of the observations x1:n, and their sampling distribution f (·|θ ) ;
m = ( f (·|θ ); θ ∈ ).