Exam Questions 2022-2023 Flashcards
(a) State the conclusion that the Dutch book argument seeks to support. [2 points, ca. 30 words]
The conclusion the Dutch book argument aims to support is that rational people must have
subjective probabilities for random events that satisfy the standard axioms of probability, namely:
- Axiom 1: All probabilities are numbers between 0 and 1.
- Axiom 2: If a proposition is certainly true, it has a probability of 1. If certainly false, it has prob. 0.
- Axiom 3: If h and h* are exclusive alternatives, then P(h or h) = P(h)+P(h)
(b) Explain how the Dutch book argument proceeds to reach its conclusion, illustrating how the argument proceeds with an example. [5 points, ca. 350 words]
The Dutch book argument starts from these two premises.
(P.1) If your degrees of belief do not conform to the rules of probability, there are possible betting situations where you are guaranteed to lose money, you fall prey of a Dutch book.
(P.2) You do not want to lose money. If premises P.1 and P.2 are true, then the conclusion follows that your beliefs should respect the axioms of probability.
Suppose that Your degree of belief that a toss of a coin will come out Heads is 0.6 (60%) and that Your degree of belief that the toss will come out Tails is also 0.6. In this situation, your beliefs violate the probability calculus, because, by Axiom 3, P(heads or tails) = 1.2. But Axiom 1 says that this cannot happen, since all probabilities are numbers between 0 and 1.
Now, suppose you are willing to take a bet on the outcome of the coin toss. Someone offers you
(i) a bet of €10 at 1.5:1 odds that the outcome will be heads, and
(ii) a bet of €10 at 1.5:1 odds that the outcome will be tails.
Because Your degree of belief that a toss of this coin will come out Heads is 0.6 (60%) and Your degree of belief a toss will come out Tails is 0.6 (60%), your subjectively fair odds for heads and for tails are both 1.5:1. So, you should be willing to accept both bets indeed.
But now you have accepted two bets each one of which pays worse than even money. After all, if the coin lands heads, you win €10 on the heads bet, but lose €15 on the tails bet; and the same happens if the coin lands tails. So, you are guaranteed to lose. You have been Dutch booked! So, if you don’t want to be Dutch booked and lose money, then your beliefs should not violate the axioms of probability.
(c) Evaluate whether or not the Dutch book argument convincingly supports its conclusion.
[3 points, ca. 200 words]
The Dutch book argument does not provide convincing support for its conclusion in my opinion.
My opinion is supported by the following reason:
Premise P.1 is likely false for many rational agents, because betting behavior doesn’t seem to be a good guide about what anyone should believe.
First, I might dislike gambling, and so I may never gamble, which means I will never fall prey of a Dutch book.
Second, even if I liked gambling, the value I give to money
does not increase linearly, and so my betting behaviour will not readily translate into subjective probabilistic beliefs.
Third, in some cases, my belief in the outcome of some event may influence the probability of the outcome itself, which means, again, that betting behaviour about some event
will not readily translate into a probability for that event.
And finally, even assuming one satisfies the axioms of probability, that’s not a sufficient condition on rational belief, since human’s are not rational to begin with.
(ai) State what the subjectivist interpretation of probability says. [1 point – ca. 20 words]
(aii) Explain in what sense the subjective interpretation is an epistemic interpretation of probability [2 points – ca. 40 words]
Subjective probability is a type of probability derived from an individual’s personal judgment or own experience about whether a specific outcome is likely to occur. It contains no formal calculations and only reflects the subject’s opinions and past experience.
Epistemic probability, is a relation between propositions: the degree to which one proposition makes another plausible. This means that all epistemic probabilities are conditional. Subjective probability is also called epistemic since there is a degree of belief in this definition is the level of confidence that the individual’s observations are at some probability to conclude a certain outcome.
(bi) In light of an example, explain what a degree of belief is. [1 point – ca. 50 words]
(bii) In light of an example, explain how somebody’s degree of belief that a certain hypothesis is true can be revealed by possible bets that person would accept or reject. [3 points – ca. 250 words]
bi) A degree of belief is a way of quantifying how confident someone is in a particular statement or belief. For example, someone might say that they have a 60% degree of belief in a particular statement, meaning that they are relatively confident in its truth. This concept is often used in the field of probability and statistics to represent the uncertainty of a given event.
bii) A person’s degree of belief in a particular hypothesis can be revealed by the bets that they are willing to accept or reject. The higher the probabilities for the bet, the higher the degree of belief is the more confident the person making the bet is that they are correct.
For example, suppose that someone believes that a certain stock will go up in value. If they are willing to bet a significant amount of money on this belief, it shows that they have a high degree of belief in its truth. On the other hand, if they are only willing to bet a small amount or are unwilling to bet at all, it shows that they have a lower degree of belief in the hypothesis. This is because the amount of money that someone is willing to bet is directly tied to the confidence they have in the outcome of an event.
By examining the bets that someone is willing to accept or reject, we can get a sense of their degree of belief in a particular hypothesis.
Explain, in light of pertinent examples, the two problems covered in our course for the subjectivist interpretation of probability. [3 points – 250 ca. words]
One of the problems with the subjectivist interpretation of probability is that it is highly dependent on the individual’s beliefs and subjective judgement, which can vary greatly from person to person. This can make it difficult to make objective and unbiased decisions based on probability.
Another problem with the subjectivist interpretation of probability is that it does not take into account the underlying reality or objective facts about a situation. This can lead to situations where an individual’s beliefs about the likelihood of an event are not accurate or reflective of the true probabilities of that event. For example, an individual may believe that a certain outcome is highly likely based on their own subjective judgement, but in reality, the objective probability of that outcome is much lower.
Overall, the subjectivist interpretation of probability has its limitations and can be problematic in certain situations. However, it is still a widely accepted and useful way of thinking about probability in many circumstances. This interpretation is in contrast to the more commonly held objective interpretation of probability, which sees probability as a objective property of events or outcomes.
(ai) State what the frequentist interpretation of probability says. [1 point – ca. 20 words]
(aii) Explain why the frequentist interpretation counts as an objective interpretation of probability. [2 points – ca. 50 words]
What is the difference between frequentist and Bayesian interpretations of probability?
ai) The probability of an outcome is the frequency with which the outcome occurs in a long sequence of trials (maximum likelihood estimation, p values etc.). Probabilities can be found by a repeatable process of repetitions.
aii) Objective probability refers to the chances or the odds that an event will occur based on the analysis of concrete measures. Frequentist probability is a way of assigning probabilities to events that take into account how often those events actually occur. Frequentist probability is sometimes also called objective probability or empirical probability.
Frequentist statistics never uses or calculates the probability of the hypothesis, while Bayesian uses probabilities of data and probabilities of both hypothesis. Frequentist methods do not demand construction of a prior and depend on the probabilities of observed and unobserved data.
(bi) State what the propensity interpretation of probability says. [1 point – ca. 20 words]
(bii) Explain whether or not, and why, Does the propensity interpretation counts as an epistemic interpretation of probability. [2 points – ca. 50 words]
bi) The propensity interpretation of probability says that the probability is thought of as a tendency of a given type of situation to yield an outcome of a certain kind, or to yield a long-run relative frequency of such an outcome.
bii) The propensity interpretation of probability is a philosophical interpretation of probability that sees probability as a measure of an inherent tendency of a given system or object to behave in a certain way. This interpretation sees probability as a property of the system or object itself, rather than as a measure of our knowledge or uncertainty about the system. As such, it can be seen as an ontological interpretation of probability, rather than an epistemic interpretation.
(ci) Explain, in light of a pertinent example, one problem for the frequentist interpretation of probability. [2 points – ca. 200 words]
(cii) Explain, in light of a pertinent example, one problem for the propensity interpretation of probability. [2 points – ca. 200 words]
ci) One problem with the frequentist interpretation of probability is that it can be difficult to determine the probability of an event if the event has never occurred or if it is impossible to repeat the experiment a large number of times.
For example, consider the situation of flipping a fair coin for the first time. According to the frequentist interpretation, the probability of the coin landing on heads is determined by the relative frequency of the coin landing on heads in a large number of trials or repetitions of the experiment.
However, since the experiment has only been performed once, it is impossible to determine the probability of the coin landing on heads based on the relative frequency of that event in a large number of trials. In this case, the frequentist interpretation of probability is unable to provide any useful information about the probability of the coin landing on heads.
The frequentist interpretation of probability relies on the assumption that probability is determined by the relative frequency of an event in a large number of trials or repetitions of an experiment. However, this assumption is not always valid, as the relative frequency of an event can vary depending on the number of trials or repetitions of the experiment.
In cases where the experiment has only been performed once or where it is impossible to repeat the experiment a large number of times, the frequentist interpretation of probability is unable to provide any useful information about the probability of the event. As such, the frequentist interpretation of probability is not always the most useful or accurate way of determining the probability of an event.
cii) One problem with the propensity interpretation of probability is that it is not always clear how to determine the probability of an event based on the inherent tendencies a given system or object. For example, consider the situation of flipping a fair coin. According to the propensity interpretation, the probability of the coin landing on heads is determined by the inherent tendency of the coin to land on heads in a given situation. However, it is not clear what this inherent tendency or disposition might be, as the coin is equally likely to land on heads or tails in any given flip. In this case, the propensity interpretation of probability is unable to provide any useful information about the probability of the coin landing on heads.
This problem arises because the propensity interpretation assumes that probability is a property of a system or object, rather than a measure of our knowledge or uncertainty about the system. However, in many cases, our knowledge and beliefs about a object can provide more useful and accurate information about the probability of an event occurring than the inherent tendencies of the object itself. As such, the propensity interpretation of probability is not always the most useful or accurate way of determining the probability of an event.
Describe the four steps we discussed in this course that are involved in Null Hypothesis Significance Testing, and illustrate each of these four steps with simple examples. [5 points, ca. 300 words]
- The first step in NHST is to formulate the null hypothesis and the alternative hypothesis. For example, if a study is examining the effect of a new drug on blood pressure, the null hypothesis might be that the drug has no effect on blood pressure, while the alternative hypothesis might be that the drug has a significant effect on blood pressure.
- Develop expectations in the form of probability distributions for possible outcomes given the truth of hypothesis. E.g., if this treatment is not effective, then when I run an experiment, such and such differences between control and treatment groups will be observed
- Gather data/observations & Evaluate to what degree observed data violate expectations. For example how many times the drug has or does not have an effect on blood pressure.
- Draw an inference from this comparison
Based on these observations we have a mean outcome and the significance level that determines the probability that I would obtain the observed data by “luck”. In the social sciences - significance level of .05.
Further, the p value, or probability value, tells you the probability that the null hypothesis is true.
Whether or not one should reject Null is determined by comparing the p-value and the significance level. If p-value is less than or equal to significance level, then: Reject Null hypothesis, otherwise cannot rule it out.
Null Hyp (p) leads one to expect a certain range of possible outcomes (if p, then q), when observed data are far outside that range (not-q), then we can reason such data would be very unlikely if Null is true.
Explain how the basic logic of Null Hypothesis Significance Testing reflects the logic of the hypothetico-deductive method. [2 points, ca. 100 words]
In both NHST and the hypothetico-deductive method, the evidence is used to evaluate the hypothesis and determine whether it is supported by the evidence.
In NHST, this is done by calculating the probability of obtaining the observed data under the null hypothesis, known as the p-value. If the p-value is below a certain threshold (usually 0.05), the null hypothesis is rejected and it is concluded that the observed data are not due to chance and that there is a real effect or relationship present.
In the hypothetico-deductive method, the evidence is used to evaluate the scientific hypothesis and determine whether it is supported by the evidence. If the evidence supports the scientific hypothesis, it is accepted as a valid explanation for the phenomenon or observation. If the evidence does not support the scientific hypothesis, it is rejected and an alternative explanation is sought.
*The hypothetico-deductive (HD) method, sometimes called the scientific method, is a cyclic pattern of reasoning and observation used to generate and test proposed explanations puzzling observations in nature.
Identify the hypothesis to be tested.
Generate predications from the hypothesis.
Use experiments to check whether predictions are correct.
If the predictions are correct, then the hypothesis is confirmed. If not, then the hypothesis is disconfirmed.*
Explain the two problems we discussed in our course for Null Hypothesis Significance Testing. [3 points – ca. 200 words]
Failure to reject the Null does not give you reason to believe that the Null is true
In an experiment, the null hypothesis and the alternative hypothesis should be carefully formulated such that one and only one of these statements is true. If the collected data supports the alternative hypothesis, then the null hypothesis can be rejected as false. However, if the data does not support the alternative hypothesis, this does not mean that the null hypothesis is true. All it means is that the null hypothesis has not been disproven—hence the term “failure to reject.” A “failure to reject” a hypothesis should not be confused with acceptance.
Choice of significance level determines the degree to which one should be willing to accept different kinds of errors
Type I error (False Positive)
Erroneously rejecting null hypothesis
Lower significance level reduces chance of type II, but raises chance of type I
Type II error (False Negative), erroneously failing to reject null hypothesis
Higher significance level reduces chance of type I, but increases chance of type II error
Significance level argued on science community and varies across the fields. In social sciences the significance level is currently 0.05.
In 2017, a group of scientists proposed to enhance the reproducibility of the social sciences by changing the p-value threshold for statistical significance from 0.05 to 0.005.
(ai) Define what a p-value is. [1 point – ca. 25 words]
(aii) Define what reproducibility means. [2 points – ca. 80 words]
(b) Explain whether or not, and why, lowering the p-value threshold will lower the rate of Type II errors in published research. [3 points - ca. 200 words]
(c) Evaluate whether and how lowering the p-value threshold would help with the “replicability crisis.” [4 points – ca. 250 words]
Ai) In null hypothesis significance testing, the p-value is used as a determining factor whether a hypothesis can be rejected or accepted. In the case where you set your p-value threshold at 0.05 and the p-value that comes out of your testing comes out to 0.04, you can reject the null hypothesis with a 95% certainty.
To simplify: If your p-value is below the threshold value, you can reject the null hypothesis.
aii) Reproducibility is the ability to obtain the same results from an experiment when it is repeated using the same methods and conditions.
Important concept: it allows other researchers to verify the findings of a particular study and build upon them in future research.
Often considered to be a fundamental principle of the scientific method, as it ensures that research results are reliable and can be trusted.
b) Lowering the p-value threshold will not necessarily lower the rate of Type II errors in published research. The p-value is a measure of statistical significance, and it indicates the probability of obtaining a given result if the null hypothesis is true. A low p-value indicates that the result is unlikely to have occurred by chance,
The rate of Type II errors is influenced by a number of factors:
the sample size,
the underlying distribution of the data, and
the power of the statistical test being used.
In some cases, lowering the p-value threshold may actually increase the rate of Type II errors. This is because a lower p-value threshold makes it more difficult to reject the null hypothesis, which means that there is a greater chance of accepting the null hypothesis even when it is false. This can increased rate of Type II errors, as more false positives are accepted as true.
Therefore, while lowering the p-value threshold can help reduce the rate of Type I errors (falsely rejecting the null hypothesis), it is not necessarily effective at reducing the rate of Type II errors.
C) Lowering the p-value threshold would not necessarily help with the “replicability crisis,” which refers to the difficulty of reproducing the results of many published scientific studies. The replicability crisis is caused by a number of factors, including poor research practices, publication bias, and the use of inadequate statistical methods.
While lowering the p-value threshold can help reduce the rate of Type I errors (falsely rejecting the null hypothesis), it is not necessarily effective at improving the replicability of research results. In some cases, lowering the p-value threshold may actually make the replicability crisis worse, as it can lead to the acceptance of false positives as true results.
To address the replicability crisis, a number of changes to scientific research practices are needed. This can include:
improving the transparency and reproducibility of research methods,
increasing the use of preregistration and peer review,
promoting a culture of collaboration and openness in scientific research.
ai) Give the formula of Bayes’s theorem. [1 point]
(aii) Explain in words what the formula of Bayes’s theorem says. [2 points – ca. 100 words]
ai) P(A|B) = P(B I A) * P(A) / P(B)
aii) Bayes’s theorem is a mathematical formula that describes the relationship between prior probabilities and observed evidence. It is often used to calculate the likelihood of an event based on its prior probability and new evidence.
Probability of P(A) is prior probability of the hypothesis: P(B|A) is called the posterior probability if the hypothesis. This is because P(A) is a rational degree of belief making the observation that is, prior to observation, while P(A I B) is our rational degree of belief after making the observation. the probability of event B occurring, given event A has occurred. P(A) – the probability of event A. P(B) – the probability of event B.
This allows us to update our probabilities based on new information, and to take into account the likelihood of different events given what we already know.
(b) Describe three advantages that the Bayesian approach has over the classical approach to statistical testing, which we discussed in our course. [4 points – ca. 300 words]
- Allows us to account for our previous
knowledge of the world (priors) , since formulated competing hypotheses have been assigned a prior probability based on previous experience. - Allows us to check how much the data confirms or disconfirms a hypothesis. Much better than just rejecting H0
Evidence confirms H if it raises confidence that it is trueP(healthy|negative)>P(healthy)Warrants higher belief that I am healthy
Evidence disconfirms H if it lowers confidence it is trueP(healthy|positive)<P(healthy)Warrants lower belief that I am healthy
- Informs us on how to adjust our beliefs in the different hypotheses
(c) Explain two problems for the Bayesian approach to statistics, which we discussed in our course. [3 points – ca. 250 words]
There are two problems for the Bayesian approach to statistics.
The first problem is how can we define priors. This is a problem because there are often no objective criteria to define priors in hypotheses which leads to confusion. This leads to confusion because different degrees of belief (or levels of truth) turn out to be warranted (justified or guaranteed) and different priors can be assigned by different people. In other words, different subjective degrees of belief in different hypotheses can lead to different conclusions.
The second problem is that Bayesianism is not always the right approach. For some fields, abductive reasoning (making a probable conclusion from what you know) seems more apt for inquiry than Bayesian statistics. Abductive reasoning is different from deductive and inductive reasoning, which are other forms of logical inference.
Abductive reasoning is often used in scientific research to generate hypotheses and develop theories to logically arrive at a conclusion, while inductive reasoning is the process of using observations to make generalizations about a larger population. In contrast, abductive reasoning is used to arrive at the best possible explanation for a given set of observations, without necessarily arriving at a definitive conclusion.
(a) Refer to the example of fracking and Earthquakes to describe three general characteristics, covered in our course, of causal reasoning in science and everyday life. [4 points – ca. 250 words]
- cas. relationships are learned based on the timing, frequency and location of events, their correlation between these can be suggestive of a casual relationship. for example location and frequency of earthquakes
- testing casual hypothesis involves doing something in the world, such as an intervention. leaving some factors unchanged and provide more insight into relationships and correlation of outcomes
- cs. reasoning has great practical significance, knowing cause — how to make things and prevent things happening
Causal reasoning in science and everyday life often involves identifying and testing potential explanations for observed phenomena. In the example of fracking and earthquakes, scientists may initially observe that areas with high levels of fracking activity experience more earthquakes than expected, and may then seek to test potential explanations for this relationship, such as the idea that fracking causes the increase in earthquakes.
Causal reasoning often involves considering multiple factors that may be contributing to the observed phenomenon. In the example of fracking and earthquakes, scientists may consider not only the direct effects of fracking on the earth’s crust, but also other factors that could be contributing to the increased earthquake activity, such as natural geological processes or changes in water levels.
Causal reasoning often involves using evidence and data to support or refute potential explanations for the observed phenomenon. In the example of fracking and earthquakes, scientists may collect data on earthquake activity in areas with and without fracking, and use this data to determine whether there is a statistically significant relationship between fracking and earthquakes. This evidence can then be used to support or refute the hypothesis that fracking causes earthquakes
(b) Describe the view that causal relationships are relationships of difference-making in
light of one example. [3 points – ca. 150 words]
The view that causal relationships are relationships of difference-making suggests that a cause must make a difference to the outcome or phenomenon being explained. In other words, if the cause were removed or changed, the outcome would be different.
For example, consider the relationship between smoking and lung cancer. The view that causal relationships are relationships of difference-making would suggest that smoking is a cause of lung cancer because it makes a difference to the likelihood of a person developing the disease. If a person were to stop smoking, their likelihood of developing lung cancer would be reduced. This difference in likelihood is a key aspect of the causal relationship between smoking and lung cancer.
(c) Define the notion of a common cause, and provide one example to illustrate it. [3 points
– ca. 100 words]
A common cause is a third factor that can explain the relationship between two variables. It is a factor that is associated with both variables and may be the underlying reason for the observed relationship between them.
For example, consider the relationship between income and education level. It may be observed that people with higher incomes tend to have higher levels of education. However, this relationship may be explained by a common cause such as access to quality education. In this case, access to quality education is a factor that is associated with both income and education level, and may be the underlying reason for the observed relationship between these variables.
(a) Describe the view that causal relationships are patterns regular association between
variables in light of one example. [3 points – ca. 150 words]
The view that causal relationships are patterns of regular association between variables suggests that when one variable consistently precedes or correlates with another variable, it may be considered a cause of that variable. For example, if researchers consistently observe that a certain medical treatment is followed by a decrease in symptoms of a particular disease, they may conclude that the treatment is a cause of the decrease in symptoms. In this case, the regular association between the treatment and the decrease in symptoms is the causal relationship.
Explain three problems for the view that causal relationships are patterns of regular
association between variables in light of pertinent examples. [4 points – ca. 300 words]
One problem with the view that causal relationships are patterns of regular association between variables is that it does not take into account the potential for confounding factors. For example, if researchers observe that a certain medical treatment is consistently followed by a decrease in symptoms of a particular disease, it is possible that the treatment is not actually causing the decrease in symptoms, but rather that some other factor is responsible. For example, the patients who receive the treatment may be more likely to recover for reasons unrelated to the treatment itself.
A second problem with this view is that it may lead to incorrect conclusions about causality. For example, if researchers observe a regular association between two variables, but do not take the time to carefully control for confounding factors and conduct further analyses, they may mistakenly conclude that there is a causal relationship when in fact there is not.
A third problem with the view that causal relationships are patterns of regular association is that it does not necessarily account for the direction of causality. For example, if researchers observe a regular association between two variables, it is possible that one variable is causing the other, or that the other variable is causing the first. Without further analysis, it is impossible to determine the direction of causality from the pattern of association alone.
State the common cause principle, and explain in what sense it is a bridge principle for causal reasoning. [3 points – ca. 100 words]
he common cause principle is a fundamental principle of causal reasoning that states that two events are causally related if they are both caused by a common third event. This principle is often referred to as the “bridge principle” because it allows us to bridge the gap between two events that may be correlated but do not have a direct causal relationship. For example, if we observe that a person’s symptoms improve after taking a certain medical treatment, and we also observe that the person’s symptoms tend to improve on their own over time, the common cause principle would suggest that the improvement in symptoms is due to the natural course of the disease rather than the treatment itself. By identifying a common cause for two events, the common cause principle helps us to understand the true nature of their relationship and avoid making incorrect causal inferences.