Probability and Uncertainty
Description
Our knowledge of a system can be limited as a result of many of the problem factors discussed previously:
- Element scope, action scope, outcome scope and relationship scope all result in structural complexity.
- Unclear causality reduces our ability to predict consequences.
- Complex behaviour makes it hard for us to see patterns or calculate behaviour. This is especially true if the problem has a lengthy time scope because the significance of system sensitivity increases the further we peer into the future.
- Novelty means that we can't rely on past behaviour to indicate what will happen in future (because the system is new or significantly changed). Even if a system isn't novel at the moment, a lengthy time scope increases the likelihood that it could change significantly and become novel in the future.
Limited knowledge of a system makes it difficult to predict the behaviour of the system. Probability and uncertainty are ways of representing our inability to predict behaviour perfectly. They allow us to make decisions even with limited knowledge, making the best of a bad situation.
Definitions
The terms that are used for types of unpredictable behaviour can be ambiguous and can mean different things in different areas of expertise. The concepts can also get complicated. To keep things as simple as possible, we'll use the following definitions.
Certainty
This is the easy case where we know exactly what will happen.
Probability
Even when we may not know exactly what will happen, sometimes we can put numbers on how likely the various possibilities are. A good example is a die.
Probability values indicate the frequency with which a particular outcome is expected to occur. The probability of rolling a four on a standard die is 1/6; i.e., on average we'd expect to roll a four once in every six throws.
How can we know what a probability value should be?
- The clearest way is to know how many ways in which the outcome could occur, and divide that by the total number of things that could happen. This is feasible when there's not too many things that could happen, such as drawing a card from a deck. However, there are a huge number of different ways to roll a four on a die. Fortunately, we can take a shortcut: because a die is symmetrical, there's an equal number of ways of rolling a one, two, three four, five or six. This means that we can know the proportion of ways of rolling a four.
- The other way to come up with a probability value is to try it. If we roll a die lots of times and keep track of what numbers come up, we'll notice that four comes up about 1/6 of the total. (The reason why we have to do it lots of times is because of statistical significance.)
Since probabilities are numbers, we can use them in various thinking processes that have mathematical underpinnings. This might sound scary but it can be easy. For example, if someone bets that you can't roll a four on a die, and offers you double your money back if you do so, you shouldn't take the bet. However, if they offer you ten times your money back, you should take the bet. Just because we can't know exactly what will happen when we roll the die doesn't mean that we can't make decisions involving it. It does, however, mean that we'll get it wrong sometimes.
When it isn't possible to come up with a probability value using either of those approaches, the value is sometimes just guessed. Two ways to do this are:
- If you think it's unlikely, pick an arbitrary low number (e.g., 0.1). If you think it's likely, pick an arbitrary high number (e.g., 0.9).
- Compare the event with a gamble that feels similar. For example, if you would take the bet only if the payout is five times the cost, the probability is about 0.2.
Guessing probabilities is tempting because it allows the values to be used in thinking processes. However, it's dangerous to use guessed values because they can be very inaccurate, which can lead to incorrect conclusions. If guessed values have to be used, sensitivity analysis should be conducted to see if the conclusion would change if the actual probability is very different to the guess.
Uncertainty
Sometimes there isn't a valid way to come up with a probability value for something that could happen. This applies to events for which we can't assess how many ways each possible outcome could happen, that don't have any history we can assess, and don't have any way for us to gather data by trying them. An example is whether aliens would attack us if we ever meet any.
Sometimes we can't even work out what the possible outcomes could be, such as the long-term health risks associated with a new drug. It's possible that the drug could result in medical problems that have never happened before. Since we don't even know what new diseases it could cause, we definitely can't work out how frequently they'd occur.
Unfortunately, the most significant system changes are the least predictable. If the changes are so large that we're effectively dealing with a different (novel) system, we may have to think in terms of uncertainty rather than probability. Uncertainty increases the further we look into the future (van der Heijden 1996, p. 92).
Making decisions involving uncertain events is harder than making decisions about probabilistic events. This is because we have less information on them. With probabilistic events, we know what could happen and how likely each outcome is. With uncertain events, we may know neither of those things. As a result, different thinking processes need to be used when uncertainty is present, and there is a greater chance of getting it wrong.
There are some approaches for attempting to assign probability values to uncertain events. The advantage of doing so is that it's easier to make decisions because we can make use of the probability values. However, the approaches are unreliable and can introduce errors. Perhaps worse, they give us a feeling that we know more than we actually do, leading to overconfidence in our decisions. Examples of approaches to assign probability values with insufficient information include:
- There's a school of thought that if you don't know what the probabilities of a set of outcomes are, you can assume that they're all the same (e.g., ‘you might as well toss a coin’). However, this is making an unjustified assumption: with no information, there's no reason to believe that there's an equal number of ways with which each outcome could occur (Spiegelhalter, p. 38).
- If you don't know a probability value, just guess. Unfortunately, guesses of probability can be wildly inaccurate. For example, at the time, professional estimates of the reliability of a Space Shuttle mission ranged from 1:100 to 1:100,000. In retrospect, 1:10 seems to be the most accurate value (Spiegelhalter, p. 137).
Risk
Risk management is a common process that is built on probability. Although risk management isn't used explicitly in the thinking framework described on this site, many of its ideas will be used.
Imagine that you inherit a building. It's in poor condition. There are two things that could go wrong with it:
- There is a 1% probability of structural failure. If that happens, it would cost $1,000,000 to rebuild.
- There is a 50% probability that the electrical system could fail. If that happens, it would cost $15,000 to repair.
Unfortunately, you can only afford to fix one of these problems. Which one should you fix?
This is where risk comes in. We know the probability and possible consequences of each of the hazards. If we multiply a hazard's probability (P) and consequences (Q) together, we get the risk (R) of that hazard; i.e., P × Q = R. In the example above, the risk of structural failure is $10,000, whereas the risk of electrical failure is $7,500.
What do those numbers mean? Risk is the average of the consequences that would be expected if the hazard existed a large number of times. For example, if there were many buildings in this condition, about 1% of them would collapse (costing $1,000,000 each) but about 99% of them would stay standing (costing $0). If we average those costs, we'd get about $10,000. Even though we only own one building, those risk values still tell us how significant each hazard is. Since the risk of structural failure is greater than the risk of electrical failure, that's the hazard that we should fix — even though it's much less likely to happen.
In the example above, each hazard (e.g., structural failure) was all-or-nothing: either the building totally collapsed or it didn't. In reality, things can be more complicated; e.g., in addition to having a 1% probability of $1,000,000 damage, there could be a 10% probability of a partial collapse incurring $200,000 damage. The overall level of risk is simply the sum of those possibilities.
Since risk is just the average value of consequences, we can also use it in a positive sense. For example, the 'risk' (or 'expected value') of winning a raffle is the probability of doing so multiplied by the prize amount. If that value is greater than the cost of a ticket, buy one.
The concept of expected value can also be applied to things that we can't easily put numbers on. For example, one of our priorities may be happiness. To compare options, we should consider both the degree of happiness, and the likelihood of achieving it, for each option. One option may have a great chance of making us pleasantly happy, while another option may have a tiny chance of making us deliriously happy. Such comparisons are harder when we can't use numbers, and we may not get the answers right all the time, but the basic idea of considering both probability and consequences remains valid.
Examples
Global Warming
How do we come up with probabilities for global warming? There have been climate changes in the past, but the current extent of human activity is new. We can't count how many ways each possible outcome could happen, and historical records may not apply because the situation is different. So, instead of looking at reality, we use mathematical or computer models to simulate the climate. If we do this lots of times using different assumptions and data, we can see how often each outcome occurs, which lets us calculate probability values.
Of course, using models instead of reality introduces a new form of error: the model won't be exactly the same as reality. Different models give different probabilities. We need to be careful not to choose to believe a particular model just because it generates results which agree with our preconceptions or priorities.
Ball Tracking
Fox Sports via Wikipedia, 2016, used under Fair Use for educational purposes.
Several sports use computer-simulated ball-tracking. In cricket, it is used to indicate whether the ball would have hit the stumps if it hadn't hit the batter first. Because information on the ball's trajectory and other factors is imperfect, it isn't possible to calculate exactly where the ball will go. However, the television graphic that presents the result of the analysis shows the ball in one specific position as it reaches the stumps.
Channel 9 via news.com.au, 2016, Used under Fair Use for educational purposes.
In fact, the position at which the ball is shown is just the centre of a range of possibilities. There is about a 50% probability that the ball could arrive at a position more than 4 mm from the indicated location. Even larger discrepancies are possible, but with lower probability. Because of this, if the ball appears to just graze the stumps, ball-tracking evidence is not used. This is often to the incredulity and disgust of commentators and players, who assume that the indicated trajectory is the exact path that the ball would have taken.
Related Issues
This section describes similarities and differences between this issue and related problem factors and thinking traps.
Complex Behaviour
Even if a system behaves consistently, complex behaviour can still make it difficult for us to work out how the system would respond to our actions.
Unclear Causality
When systems don't behave consistently but possess variability, it is more difficult to determine causality.
Overconfidence
Overconfidence can cause us to assume that our knowledge or effectiveness are greater than they actually are, leading us to treat things as certain rather than subject to probability.
A more subtle consequence of overconfidence is a willingness to assign probability values to uncertain events when we don't know enough to do so accurately.
Intuition
Intuition is able to make judgements despite having imperfect knowledge, so it can deal with probability and uncertainty to some extent. However, it is susceptible to some types of mistakes when it does so. For example, intuition tends to misinterpret random coincidences caused by probability to be causal relationships.
Can misinterpret random coincidences caused by probability to be causal relationships. unclear-causality qv
If the lack of information about uncertain events precludes reasoning, intuition may be required as a last resort.
Risk Factor Misjudgement
Without adequate information, estimating probabilities is difficult. We are susceptible to various kinds of misjudgement. The most obvious trap is to think in back-and-white terms (it will/won't happen) rather than shades of grey (it might happen).
Familiarity Bias
The need to think in terms of probabilities is often indicated by variations in a system's behaviour. If we don't take a sufficiently long view of the system's behaviour, we may not notice variability and may conclude that the system's behaviour is constant or is just changing consistently. An example is climate change: if we just consider the current year, or the last couple of years, we won't see the complicated variations that occur over longer timescales.
The potential to fall into this trap is greater for problems that have a long time-scope.
No Good Options
If the problem system's behaviour exhibits probability or uncertainty, we can't be sure whether anything we do would actually fix the problem. We might even make the problem worse. This could lead us to feel that there are no good options for dealing with the problem, leading to a temptation to do nothing about it.
This concern is especially acute if the action we take could accidentally lead to bad consequences that subsequent adaptation couldn't fix.
Related Engine Processes
Later, we will describe a thinking framework that comprises multiple processes. This section points forward to the processes that deal with this page's topic.
Problem Assessment
We may use problem assessment to determine which problem to solve. The importance of a problem should take into account the level of risk it poses; i.e., the product of probability and consequences.
Scenario-Based Planning
Scenario-based planning can help us deal with problems that are subject to uncertainty by considering multiple diverse ways in which the future could develop. It isn't necessary to be able to assign probability values to each possible scenario.
Systems Thinking
Systems thinking can be used to assess how our actions can influence a problem system. The system will contain multiple inter-related elements. Systems thinking doesn't require us to be able to put precise numbers on our actions, or on the elements and relationships within the system. If we express those things in terms of probabilities, systems thinking can indicate the resulting system behaviour in terms of probabilities.
As an example, we don't know exactly how much carbon dioxide we will emit in the next few years, but we can use probability values to indicate the range of possibilities. When we do so, a system model can tell us the amount of global warming that would result, in terms of probability values indicating the range of possibilities.
Optimisation
As with Systems Thinking (above), optimisation is possible when system element values involve probabilities. However, optimisation can't guarantee that the best result will always be obtained in reality; it can only balance the range of possibilities.
Statistical Significance
Statistical significance can indicate whether an observation is meaningful of whether it's just a random blip caused by probability. For example, because the climate system is complex, it appears to behave in terms of probabilities. Because of this, it's possible that there could be a sequence of relatively cold years. Statistical significance can be used to assess whether such a sequence means that global warming isn't happening, or whether it's just random variations.
Sensitivity Analysis
Sensitivity analysis lets us see how a system's outcome would change if something else in the system changed (such as an action, element or relationship value). This is especially important when we have to specify things in terms of probabilities. For example, we might hope to reduce greenhouse gas emissions by 50%, which would result in global warming of 1.5℃. However, we can't be certain of achieving a 50% reduction; there is a probability that we might only achieve a 25% reduction. Sensitivity analysis would let us see the effect of that variation on the level of global warming that would result. This gives us a better idea of the range of possible outcomes that could occur, considering our limited knowledge of our future actions and other factors.
If a system element or relationship is subject to uncertainty, we might estimate probability values using judgement (i.e., guessing). Because such estimates can be very inaccurate, sensitivity analysis should be used to assess the significance of possible errors in our guesses.
Project Management
Risk management implies that problems that pose the highest risk should be accorded the highest priority. However, this doesn't mean that actions to fix those problems should always be implemented first. We may be able to defer action on a high-risk problem without significantly increasing the level of risk that it poses. This could buy us time and/or resources to allow us to fix a lower-risk problem that can't wait. We need to distinguish between what is urgent and what is important.
Project management can be used to determine the sequencing of actions when resources are constrained (i.e., when we can't do everything at once). While the primary focus of project management is on task scheduling, it can also deal with differing task significance, such as risk.
When a problem is subject to probability or uncertainty, it can be beneficial for us to monitor the system's behaviour for a while before taking action to fix the problem. Monitoring increases our knowledge, which can reduce the magnitude of the probability or uncertainty, allowing us to make better decisions about the actions we should take. Project management can include tasks for monitoring system behaviour.
Adaptation
When a problem is subject to probability and/or uncertainty, our initial assessment of the actions required to fix a problem could be less effective than we expected. This makes it important to monitor the system and adapt as we learn more about how it responds.
The variability in system behaviour caused by probability/uncertainty complicates our attempts to assess the effectiveness of our actions, since there may not be an obvious correlation between what we do and how the system responds.
Key Points
Probability and uncertainty can happen. When they do, we should avoid black-and-white thinking.
We can still think and make decisions when probability and uncertainty are present.
Further Reading
To Do
Murdered Darlings
There is a school of thought that all forms of probability arise from lack of knowledge. In that case, probability is a thinking trap rather than anything inherent in the problem's system. No: the 'trap' is ignoring probability (esp. B&W as per risk-factor-misjudgement). Most/all problem factors are only issues when compared to human abilities (eg, *-scope): nature doesn't get overwhelmed, but people might.
Thinking and decision-making are simplest when the problem system's behaviour is perfectly predictable. That makes it relatively easy to work out how the system would respond to our actions. However, many systems don't behave in a consistent manner, so we can't predict the effects of our actions with great confidence.
Probability is an assessment of uncertainty (Chivers 124-125). We'll go further and say that probability is a quantified measure of uncertainty.
Variability: system outputs are not constant. Unimportant; only unpredictability matters. Apparent variability may be only a consequence of inability to predict perfectly (die, weather). If we can assess variability, we can assign probability.
A novel system will not nec behave with variability (it could be a constant but unpredictable value); we have no knowledge of how it will behave. Novelty could mean that we don't know the scope, causality and/or behaviour of the system adequately. Therefore the consequences of any or all of those issues could apply to the system, but we don't
know.
'Inconsistency' = 'variability' but with pejorative connotations.
Unpredictability: consequence of variability if we can't work out how to determine what the future variations will be (eg, due to lack of pattern, historical data, analytical way to determine relative number of ways with which each outcome could occur).
Uncertainty increases (i.e., the ability to quantify probability decreases) the further we look into the future: bad for GW! (van der Heijden 1996, p. 92).
Just use RM to prioritise events (effects), or can it be used to prioritise treatments (actions)? Perhaps the latter is best left to optimisation qv. Optimisation's approach is probably more general than RM, which is a simple subset (can I prove this??). Simple RM is just a way of prioritising problems that are subject to P; optimisation does this (and more).
Link this to limited link-depth-following in thing-scope qv?
Rename? unpredictability?