Statistics

Question 1

What is Bayesianism?

Answer 1

Bayesian statistics can evaluate a potentially large set of competing hypotheses. Bayesian statistics interprets probabilities as “degrees of belief” or “subjective confidence in hypotheses”. Probabilities are assigned to hypotheses.

Question 2

Which one(s) of these assign probabilities to the hypotheses: Fisher null-hypothesis testing, Neyman Pearson Decision theory, Bayesian hypothesis testing?

Answer 2

Bayesian.

Question 3

How is Fisher’s Null Hypothesis Testing performed?

Answer 3

• Specify the main hypothesis H
• Devise an experiment to test H and specify its possible outcomes (the so called “test statistic”).
• Determine the distribution of the test statistic, given that H is true
• Observe the experimental outcome
• Calculate the p-value. This is the probability of observing a result at least as extreme as the one observed, given that the hypothesis is true.
• If the P-value is smaller than a conventionally set significance level, (typically 0.05), reject H

Question 4

How is Neyman Pearson Decision theory performed?

Answer 4

Choose error rates for type I (rejecting a true hypothesis-false positive) and type II (accepting a false hypothesis - false negative) errors. This is the same as comparing the result against a significance level (alfa1 and alfa2)

Question 5

Name two reasons for why it is often good practice to evaluate hypotheses using statistical tools.

Answer 5

One is that some hypotheses are probabilistic. Such hypotheses have distributions, rather than individual instances, as expected observable outcomes. When we make empirical observations however, we can only observe individual instances, not distributions. Therefore, statistical measures are required in order to link multiple individual observations to the relevant distributions.
Another reason is that observations, used to test hypotheses, can be erroneous, i.e. observations and measurements are prone to random errors. Such errors can lead us to wrongly reject a correct hypothesis, or wrongly accept an incorrect hypothesis. Statistical tools can be employed to quantify error.

Question 6

What is a p-value?

Answer 6

It is the probability of observing a result at least as extreme as the one actually observed.