 So as you can see here that Chris finished this photograph and I thought that when we change chairs you had time to resolve this difficult decision problem. What is more beneficial to have some food and drink or to read the book by Ian Jordan on decision and uncertainty. So I continue and I present two other cases of decision analysis. Posteria and preposteria and I also give again very brief definition of value information in the context of preposteria analysis. So next type of analysis, this is posterior analysis and the difference between posterior analysis and prior analysis presented by Chris is that in this case the decision maker gets new information from experiment or inspection but the decision about whether to carry out this inspection is outside the decision process. So the decision maker gets the information but he is not involved in decision about the experiment. So in this case only difference that using this in new information or outcome of experiment which is denoted here as ZK, he can up here or she can update this prior probabilities and find posterior probabilities. So notation here that prior probabilities we denote as P prime and posterior probabilities is P double prime and they're conditional on this new information. In order to do that we use this bias theorem and this is very straightforward. Additional information which should be known in this case the so-called likelihood functions which describe the probability of getting this experiment outcome given certain the state of, given the certain state of the structure which we can see. So after that the decision process is very similar. First we calculate expected utilities for all possible actions and in this case the expectation operation is carried out with respect to posterior probabilities not prior probabilities. So we get these expected utilities for all possible actions and then we should select the action A asterisk which leads to the maximum utility and this is the end of the decision process. So once again I would like to show this full tree because preposterior analysis is multi-stage decision analysis. So in this case the decision maker should make two decisions first to decide if to carry out experiment or inspection and what type of inspection to carry out and then about actions which should be undertaken for example maintenance actions. So this preposterior analysis can be done in two forms extensive form and normal form. So I'll start with explanation about this extensive form of analysis. So in state extensive form of analysis starts with assumption that inspection and its outcome unknown. So it basically backward analysis. So we start from the right end of the decision tree and then we go back to its starting point. So first we assume that experiment or inspection and its outcome unknown and by this we reduce the problem to asterial analysis. So for given inspection and its outcome we need to find the maximum expected utility. So once again based on this assumed information which we know we can update prior probabilities for the structural states and then for each combination of inspection and its outcome we calculate maximum expected utilities. So it's exactly like in posterior analysis. But the difference of course that we don't know what inspection has been selected and what has been its outcome. So outcomes that they're once again uncertain parameter or random variable. So once again we need to apply the expectation operation in order to calculate expected utilities for different inspections. And for that we once again need some new information which is represented by probabilities of experiment outcomes that. So for each experiment there can be obviously different probabilities. So after we formulated these probabilities we carry out this expectation operation with respect to these probabilities and we get expected probabilities for each possible inspection. Next step obviously once again to select the maximum expected utility and the experiment or inspection which corresponds to it. And this is the last step. So this is the way how this extensive form of pre-pastry analysis is carried out. Now we can see this normal form and in this case the analysis is carried out forward from the starting point on the left hand side or the tree to its right end. And the first step here is to formulate is this so-called decision rule. So basically we need to assign optimal action A to each possible outcomes of our inspection. And if you think about it it's quite similar to the first step of extensive form of pre-pastry analysis. We just don't need to calculate utilities. Basically we have all possible outcomes and for each possible outcome we should select action which would lead to the maximum utility. So that's the way how to reset this decision rule. And after that we can carry out analysis taking into account now that instead of actions we consider decisions which are related to inspection outcomes. And information which we need for this analysis are exactly the same as for pastry analysis. We need to know prior distributions of the state of the structure P prime theta. And we need to know likelihoods or probabilities of observing some experimental outcomes given a certain state of the structure. The difference is that we need these likelihoods for each inspection possible inspection type. So after that we need to calculate expected utility for each combination of inspection, selected inspection and decision D. And then from all calculated utilities for these strategies or combinations we select the maximum one and the strategy ED inspection plus this decision gives us the most optimal decision. So these are two forms. According to Rafer and Schleifer for the evaluation of value of information extensive form is most suitable. And concerning normal form for example in one of his publications Michael Farber showed that it can be quite useful then we consider risk-based inspection and maintenance planning. So now value of information. So one of the main applications of pre-pastry analysis is the calculation of the value of information. And first we define the value of information for particular outcome or for particular piece information Z. So as you have seen in this extensive form of pre-pastry analysis for each actions and the states of the structure we calculated this maximum expected utilities US risk. So these are maximum expected utilities for any combination of E and experiment it's outcome Z. So now if we consider for the outcome Z the difference between this utility and the maximum utility which we can calculate by prior analysis without any experiments and outcomes this will represent the value of information for this particular piece of information Z. The other definition more general definition was already presented by Henning in terms of benefits. Here I present the same in terms of utilities. So basically if we denote U not asterisk the maximum utility obtained by prior analysis and by U one asterisk the maximum utility obtained by pre-pastry analysis. In this case it's in extensive form formulation. The difference between these two utilities will give us the value of information. This is also called expected value of sample information because we take into account that at the time we estimate it we don't know nothing about experiment it's outcomes and the state of the structure. There are other definitions of value information so conditional value of sample information it's very similar we know the only difference that it's related to the citation when we estimate the information after the experiment has been carried out and we know it's outcome so it's conditional value of sample information. There is also such notion as perfect information the idea of perfect information that this information enables us to establish the exact state condition of the structure. So and for perfect information we can also calculate its expected value and conditional value and these values for perfect information they're basically upper limits for our value of information and that's it.