This chapter provides a brief presentation of the philosophical and math¬ematical foundations of Bayesian inference. The connections to classical statistical inference are also briefly discussed.

Philosophy of Bayesian inference

The purpose of Bayesian inference (Bernardo and Smith, 1994; Gelman et al., 2004) is to provide a mathematical machinery that can be used for modeling systems, where the uncertainties of the system are taken into account and the decisions are made according to rational principles. The tools of this machinery are the probability distributions and the rules of probability calculus.

If we compare the so-called frequentist philosophy of statistical analysis to Bayesian inference the difference is that in Bayesian inference the prob¬ability of an event does not mean the proportion of the event in an infinite number of trials, but the uncertainty of the event in a single trial. Because models in Bayesian inference are formulated in terms of probability dis¬tributions, the probability axioms and computation rules of the probability theory (see, e.g., Shiryaev, 1996) also apply in Bayesian inference.

Connection to maximum likelihood estimation

Consider a situation where we know the conditional distribution p(yk | θ) of conditionally independent random variables (measurements) y1:T = {y1,…, yT}, but the parameter θ ∊ ℝd is unknown.

The aim of this book is to give a concise introduction to non-linear Kalman filtering and smoothing, particle filtering and smoothing, and to the related parameter estimation methods. Although the book is intended to be an introduction, the mathematical ideas behind all the methods are carefully explained, and a mathematically inclined reader can get quite a deep understanding of the methods by reading the book. The book is purposely kept short for quick reading.

The book is mainly intended for advanced undergraduate and graduate students in applied mathematics and computer science. However, the book is suitable also for researchers and practitioners (engineers) who need a concise introduction to the topic on a level that enables them to implement or use the methods. The assumed background is linear algebra, vector calculus, Bayesian inference, and MATLAB® programming skills.

As implied by the title, the mathematical treatment of the models and algorithms in this book is Bayesian, which means that all the results are treated as being approximations to certain probability distributions or their parameters. Probability distributions are used both to represent uncertainties in the models and for modeling the physical randomness. The theories of non-linear filtering, smoothing, and parameter estimation are formulated in terms of Bayesian inference, and both the classical and recent algorithms are derived using the same Bayesian notation and formalism. This Bayesian approach to the topic is far from new. It was pioneered by Stratonovich in the 1950s and 1960s – even before Kalman's seminal article in 1960.

Part VI discusses queueing analysis where the arrival process and/or service process are generally distributed.

We start with Chapter 20, where we study empirical job size distributions from computing workloads. These are often characterized by heavy tails, very high variance, and decreasing failure rate. Importantly, these are very different from the Markovian (Exponential) distributions that have enabled the Markov-chain-based analysis that we have done so far.

New distributions require new analysis techniques. The first of these, the method of phase-type distributions, is introduced in Chapter 21. Phase-type distributions allow us to represent general distributions as mixtures of Exponential distributions. This in turn enables the modeling of systems involving general distributions using Markov chains. However, the resulting Markov chains are very different from what we have seen before and often have no simple solution. We introduce matrix-analytic techniques for solving these chains numerically. Matrix-analytic techniques are very powerful. They are efficient and highly accurate. Unfortunately, they are still numerical techniques, meaning that they can only solve “instances” of the problem, rather than solving the problem symbolically in terms of the input variables.

In Chapter 22 we consider a new setting: networks of Processor-Sharing (PS) servers with generally distributed job sizes. These represent networks of computers, where each computer time-shares among several jobs. We again exploit the idea of phasetype distributions to analyze these networks, proving the BCMP product form theorem for networks with PS servers. The BCMP theorem provides a simple closed-form solution for a very broad class of networks of PS servers.

In Chapter 21, we saw one application for phase-type (PH) distributions: If we need to analyze a system whose workload involves distributions that are non-Exponential (e.g., high-variability workloads), then we can use a PH distribution to at least match 2 or 3 moments of that workload distribution. This allows us to represent the system via a Markov chain, which we can often solve via matrix-analytic methods.

In this chapter we see another application of PH distributions. Here, we are interested in analyzing networks of Processor-Sharing (time-sharing) servers (a.k.a. PS servers). It will turn out that networks of PS servers exhibit product form solutions, even under general service times. This is in contrast to networks of FCFS servers, which require Exponential service times. Our proof of this PS result will rely on phase-type distributions. This result is part of the famous BCMP theorem [16].

Review of Product Form Networks

So far we have seen that all of the following networks have product form:

• Open Jackson networks: These assume probabilistic routing, FCFS servers with Exponential service rates, Poisson arrivals, and unbounded queues.

• Open classed Jackson networks: These are Jackson networks, where the outside arrival rates and routing probabilities can depend on the “class” of the job.

• Closed Jackson networks

• Closed classed Jackson networks

We have also seen (see Exercise 19.3) that Jackson networks with load-dependent service rates have product form. Here the service rate can depend on the number of jobs at the server. This is useful for modeling the effects of parallel processing.