K. Nishijima. Definition and use of Bayesian probabilistic networks 1/32

Transcription

1 The Probabilistic Analysis of Systems in Engineering 1/32 Bayesian probabilistic bili networks Definition and use of Bayesian probabilistic networks K. Nishijima

2 2/32 Today s agenda Probabilistic representation of systems Representation of systems with (random) variables Observable and unobservable variables joint probability, conditional probability, marginal probability, prior/posterior probability, predictive probability Bayesian probabilistic networks (BPNs) Graphical representation / conditional probability (tables) Advantages of BPNs Example

3 3/32 Representation of systems Systems may be characterized by variables and the systems performance may be described by using the variables. Model x1, x2,..., xn f ( x, x,..., x ) i 1 2 n

4 4/32 Representation of systems Systems may be characterized by variables and the systems performance may be described by using the variables. Ship hull structure system Corresponding model Variables: Self weight Wave loads Thickness of plates Stiffness Corrosion, etc.

5 5/32 Representation of systems If the states of the systems are uncertain, the systems may be characterized by random variables. Model Probabilistic model x1, x2,..., xn X 1, X 2,...,, X n f ( x, x,..., x ) fi( X1, X2,..., Xn) i 1 2 n

6 6/32 Task in modeling The main task in modeling is to identify the interrelations between all the variables involved in the models in logical and/or probabilistic manners. Probabilistic model X 1, X 2,...,, X n f ( X, X,..., X ) i 1 2 n Example of logical relation: X3 = X1+ X2 Example of probabilistic relation: PX [ X, X ] (The logical relation is a special case of the probabilistic relation.)

7 7/32 Aim of analysis The aim of the analysis with models is to evaluate the quantities of interest, e.g. the probability of the occurrence of critical situations, using the logical and probabilistic relations in the models. Probabilistic bili model Example of quantities of interest: t X1, X2,..., Xn P F where = P [ X = 1] f i ( X 1, X 2,..., Xn ) X = I X1 < X2 [ ]

8 8/32 Observable and unobservable Observable variables The values of some of variables can be observed by e.g. measurements. An example is the annual maximum wind speed at a given location: 10 m/s, 12 m/s, 29 m/s, 14 m/s, 35 m/s, 18 m/s.

9 9/32 Observable and unobservable Unobservable variables (latent variables, model parameters, etc.) Unobservable variables are not directly observed but rather inferred from the variables that are directly measured. An example is the mean value of the annual maximum wind speed at one location. The mean value may be inferred from the observations as: ( )/6 = m/s

10 10/32 Observable and unobservable Example The annual maximum wind speeds X i (i=1,2,3,4,5) are assumed modeled as: Xi iid N μ σ 2 (, ) Here, X i s are the examples of observable variables, and μ and σ 2 are the examples of unobservable variables. In Bayesian approaches, μ and σ 2 themselves may be considered as random variables.

11 11/32 Joint probability Any models that represent systems can be fully described by joint probabilities (probability densities or their combination): P[ X1, X2,..., X n ] Only few parametric families of multivariate distributions are known, e.g., multivariate Normal distribution, Dirichlet distribution. They are often less flexible; consider the case X 1 is continuous and X 2 is discrete. Quite lots of information are required for characterizing the multivariate distributions; (n 2 + n) values are required for the multivariate Normal distribution. Also, think of a multivariate discrete distribution!

12 12/32 Conditional probability An efficient (compact) way of characterizing joint probabilities is to employ conditional probabilities. Any joint probability can be built up with conditional probabilities: n P[ X1, X2,..., X ] = P[ X pa( X )] n i i i=11 where pa(x i ) represents the parents of X i (which will be later explained in the context of Bayesian probabilistic networks.)

13 13/32 Conditional probability Example Assume two situations where the joint probability of discrete random variables X 1, X 2 and X 3, each of which can take 5 possible values, are modeled in two different ways: P[ X1, X2, X3] This direct representation ti requires 5 3 =125 values to describe the joint probability. PX [, X, X ] = PX [ X ] PX [ X ] PX [ ] This conditional independence representation requires only =55 values, although h this representation is less flexible than the above one.

14 14/32 Marginal probability Any marginal probability can be calculated from joint probabilities for discrete cases as: PX [ ] PX [, X.., X] i = 1 2 i 1 i+ 1 n 1 2 n For continuous cases the summations may be replaced by appropriate integrals.

15 15/32 Marginal probability Example Consider a Bernoulli trial throwing a coin 10 times. However, the probability that a head comes out is not known. Assume that X 1 represents the number that the head comes out in the 10 trials, and X 2 represents the probability that the head comes out in each trial and follows the uniform distribution [0,1]. Calculate P[X 1 ].

16 16/32 Marginal probability Example (solution) Joint probability P[ X = i, X = p] = P[ X = i X = p] P[ X = p] Marginal probability 10 i 10 i 10 = p (1 p) 1 i i i PX [ = = = = = 1 i] PX [ 1 ix, 2 pdp ] p (1 p ) dp ! i!(10 i) 1 = = (10 i)! i! 11! 11 i

17 17/32 Prior/posterior probability According to the Bayes theorem the following relation holds: PX [ 1 = x1 X2 ] P[ X2 X1 = x1] = P[ X2] PX [ = x] 1 1 Consider special cases where X 2 represents the parameter of the probability of X 1. In such cases, the relation above provides a rationale how the probability of X 2 may be updated with the observed data (X 1 =x 1 ). Thereby, P[ X ] : prior probability (distribution) 2 PX [ X = x ] : posterior probability bili (distribution). ib i 2 1 1

18 18/32 Prior/posterior probability Example Consider again the Bernoulli trial throwing a coin. Assume that 5 heads came out after 10 trials, i.e., X 1 =5. Calculate P[X 2 X 1 =5], the posterior probability of X 2.

19 19/32 Prior/posterior probability Example Posterior probability PX [ = 5 X = p] PX p X PX p 1 2 [ 2 = 1 = 5] = [ 2 = ] PX [ 1 = 5] Thus, p (1 p ) 5 5 Γ(12) P[ X2 = p X1 = 5] = p (1 p) Γ(6) Γ (6) 5 5

20 20/32 Predictive probability Predictive probability is an alias of the marginal probability when the joint probabilities are marginalized in regards to the parameters of the probabilities: P [ X ] = P [ X, X ] = P [ X X ] P [ X ] where X 2 is the parameter of the probability of X 1. P [ X ] : Predictive probability 1

21 21/32 Predictive probability Example Consider again the Bernoulli trial throwing the coin. Calculate the predictive probability P[X 1 ], using the updated (posterior) probability of X 2.

22 22/32 Predictive probability Example (solution) 1 PX [ = i ] = PX [ = ix, = pdp ] = P [ X = i X = p ] PX [ = p ] dp 1 10 Γ(12) = 0 i Γ(6) Γ(6) i 10 i 5 5 p (1 p) p (1 p) dp 10 Γ(12) 1 i i = p (1 p ) dp i (6) (6) 0 Γ Γ = Γ (11) Γ (12) Γ ( i + 6) Γ (16 i ) Γ ( i + 1) Γ (11 i) Γ (6) Γ (6) Γ (22)

23 23/32 Predictive probability Example (solution) 0.2 Predictive with prior Predictive with posterior babilit ty Pro Number of heads

24 24/32 Bayesian probabilistic networks DAG (Directed Acyclic Graph) Conditional probability (tables)

25 25/32 Bayesian probabilistic networks Example of DAG (Directed Acyclic Graph) X 1 X 2 P[ X X ] P[ X ] 2 1 1

26 26/32 Bayesian probabilistic networks Example of DAG (Directed Acyclic Graph) X 1 X 2 X 3 P[ X3 X1, X2] P[ X1] P[ X2]

27 27/32 Bayesian probabilistic networks Example of DAG (Directed Acyclic Graph) X 1 X 2 X 3 P[ X2 X1] P[ X3 X1] P[ X1]

28 28/32 Bayesian probabilistic networks Example of DAG (Directed Acyclic Graph) X 1 X 2 X 3 X 4 P[ X4 X3] PX [ 3 X1, X2] PX [ 1] PX [ 2]

29 29/32 Bayesian probabilistic networks Counter-example of DAG (Directed Acyclic Graph) X 1 X 2 X 3

30 30/32 Why are BPNs useful? Easy way of mind mapping. Platform for integrating all relevant aspects. Generic algorithms as well as software tools are available for - creating BPNs - calculating probabilities of interest.

31 31/32 Bayesian probabilistic networks Example Consider the Bernoulli trial throwing a coin 10 times. X1_before: number of heads at first trial X1_after: number of heads at second trial X2: the probability that a head comes out Will be demonstrated.

32 32/32 Modeling with BPNs Bottom-up approach The components of a BPN are built up by expert knowledge and analyses and are integrated into the BPN. Top-down approach The interrelation between the components in a BPN are fixed, but the probabilistic (quantitative) relations are estimated from observed data. Let s see these approaches with application examples later.