An experiment was carried out with the aim of updating information of a parameter θ. After the study for values of θ = (3.5, 4.0, 4.5, 5.0) corresponding values of the parameter were obtained as θ = (1.2, 1.4, 1.6, 1.8). The available past information stated that θ was uniformly distributed taking the value θ = 0.45. Test the hypothesis that θ = 4.5 against θ Not equal to 4 .5
The major problem with the video is that it brings a software when the explanation was going smoothly. Why can't you solve the problem manually with the details? It is easy to explain: Calculate the probability of observing the data under the null ( 50) using the T-distribution and take the ratio, using the posterior distribution of the mean. So some explanation of the prior and posterior distribution is required.
Thanks for taking the time to leave your comment! I agree...conceptually...that it *should* be this simple. The issue is that calculating the marginal likelihood (i.e., p(data | H1)) is not easy at all. In this case, it requires integrating the likelihood over the prior distribution, which almost always requires a software solution (because the integrals rarely admit closed form solutions...though when they do, it's very nice!). And, because this is an introductory video in the context of a course and book where the software (PsyStat app) has been used throughout, using it for these Bayesian tests is (I think) a natural thing to do.
@@TomFaulkenberry I agree that software are required to do the marginals. What I personally like is to bring the theory and derivations fully up to the point of numerical calculation and then leave the finer details of the calculations to the software. I understand that it is an introductory course. Thanks for your reply.
Can you explain the prior probability distributions for H0 and H1, since while H0 is delta == 50, the H1 is delta 50? I can see the prior distribution for H0 could be a normal distribution with mean = 50 and sd = something really tight. But I don’t know about the prior for H1. Perhaps I am overthinking….
very deep question...thanks! The idea of using Bayes factors to support a hypothesis (rather than rejecting) is typically seen as counter to a Popperian view of science (the falsification paradigm). Gelman's (2011) paper "Induction and deduction in Bayesian data analysis" is a good read on this...
@@TomFaulkenberry Thank you. As an outsider (I am definitely not a statistician) I started to realize that the Bayesian approach *seemed* counter to the Popperian paradigm, but haven't seen it discussed. I'll definitely give that paper a read.
I have a question: how do we know the sample size is big enough to calculate the BF and draw a conclusion based on that? If H0 is valid, shall we collect more data, and finally we can see the opposite? When can we be sure that the data is big enough for the hypothesis test? Thanks!
For z-test, the answer is "yes", because the z-test and t-test are really the same thing (the only difference is that a z-test uses a known value for SD, whereas the t-test uses an estimated value...for the Bayes factor, you can input the z-score in as the t-score). For proportions, you could use a normal approximation. While there are some technical concerns about using t-tests in this case, it is unclear what effect this has on Bayes factors, so this might be a good research question!
in case anyone takes issues with my saying that z and t are the "same thing", I'm talking about the form of the test statistic, not the assumptions of the test, etc. For the Bayes factor calculation context, the form of the test statistic is what matters.
The video should have explained calculation method of Posterior probability for H1, as the fig indicates it to be a cummulative probablity. The P(H1/data) should always be more than P(H0/data) , as alternative hypothesis curve is based on the sample data.
the posterior probability for H1 can be obtained from the Bayes factor for H1 over H0 as follows: BF_10 / (1 + BF_10). However, it is NOT always true that P(H1 | data) > P(H0 | data) -- for example, whenever BF_10 < 1; in this case, the data are evidential for H0, not H1.
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You're a great professor. I can tell you really care about the work you're doing. Thank you for that! Students can feel when a professor truly cares.
Thank you for this. No one else could explain it so clearly and understandable.
thanks so much!
This video was great. Thanks for such a clear explanation!
I'm glad you enjoyed it...thanks for your feedback!
An experiment was carried out with the aim of updating information of a parameter
θ. After the study for values of θ = (3.5, 4.0, 4.5, 5.0) corresponding values of the
parameter were obtained as θ = (1.2, 1.4, 1.6, 1.8). The available past information
stated that θ was uniformly distributed taking the value θ = 0.45. Test the hypothesis
that θ = 4.5 against θ
Not equal to 4
.5
Help on this prof
Thank you very much Tom. Wonderful. The best video to understand between Bayesian and Frequentist.
thanks, very useful for my thesis
really helpful, thank you!!
wow,like a spring breeze～Thank you very much
Great video Prof. I now have a clue on the topic
Very helpful example for me. Wouldn’t it be helpful to add the corresponding p-value calculation to make it easier for frequentists to accept?
As a PhD student, this is the best video that explains the concept very clearly! Thank you professor!
So incredibly well explained!
Thanks,
Excellent lecture. Wow. Thank You
Fantastic lecture
Thank you!!
Loved the video except from the arsenal banner above the blackboard... Overall still great video though haha
The major problem with the video is that it brings a software when the explanation was going smoothly. Why can't you solve the problem manually with the details? It is easy to explain: Calculate the probability of observing the data under the null ( 50) using the T-distribution and take the ratio, using the posterior distribution of the mean. So some explanation of the prior and posterior distribution is required.
Thanks for taking the time to leave your comment! I agree...conceptually...that it *should* be this simple. The issue is that calculating the marginal likelihood (i.e., p(data | H1)) is not easy at all. In this case, it requires integrating the likelihood over the prior distribution, which almost always requires a software solution (because the integrals rarely admit closed form solutions...though when they do, it's very nice!). And, because this is an introductory video in the context of a course and book where the software (PsyStat app) has been used throughout, using it for these Bayesian tests is (I think) a natural thing to do.
@@TomFaulkenberry I agree that software are required to do the marginals. What I personally like is to bring the theory and derivations fully up to the point of numerical calculation and then leave the finer details of the calculations to the software. I understand that it is an introductory course. Thanks for your reply.
Can you explain the prior probability distributions for H0 and H1, since while H0 is delta == 50, the H1 is delta 50? I can see the prior distribution for H0 could be a normal distribution with mean = 50 and sd = something really tight. But I don’t know about the prior for H1. Perhaps I am overthinking….
Doesn’t the bayes factor also depend on the power for H1? You need to specify a predicted effect size for H1, right? Or some kind of prior
Question: do you run into problems with the falsification paradigm when you say that evidence supports the alternative hypothesis?
very deep question...thanks! The idea of using Bayes factors to support a hypothesis (rather than rejecting) is typically seen as counter to a Popperian view of science (the falsification paradigm). Gelman's (2011) paper "Induction and deduction in Bayesian data analysis" is a good read on this...
@@TomFaulkenberry Thank you. As an outsider (I am definitely not a statistician) I started to realize that the Bayesian approach *seemed* counter to the Popperian paradigm, but haven't seen it discussed. I'll definitely give that paper a read.
Could you show the formulas for converting t-stats to bayes factor, or are there library functions in Python or R?
Thank you!
I have a question: how do we know the sample size is big enough to calculate the BF and draw a conclusion based on that? If H0 is valid, shall we collect more data, and finally we can see the opposite? When can we be sure that the data is big enough for the hypothesis test? Thanks!
Is this applicable for z-test or test with proportions as well?
For z-test, the answer is "yes", because the z-test and t-test are really the same thing (the only difference is that a z-test uses a known value for SD, whereas the t-test uses an estimated value...for the Bayes factor, you can input the z-score in as the t-score). For proportions, you could use a normal approximation. While there are some technical concerns about using t-tests in this case, it is unclear what effect this has on Bayes factors, so this might be a good research question!
in case anyone takes issues with my saying that z and t are the "same thing", I'm talking about the form of the test statistic, not the assumptions of the test, etc. For the Bayes factor calculation context, the form of the test statistic is what matters.
@@TomFaulkenberry really appreciate the reply and the videos that you've created
The video should have explained calculation method of Posterior probability for H1, as the fig indicates it to be a cummulative probablity. The P(H1/data) should always be more than P(H0/data) , as alternative hypothesis curve is based on the sample data.
the posterior probability for H1 can be obtained from the Bayes factor for H1 over H0 as follows: BF_10 / (1 + BF_10). However, it is NOT always true that P(H1 | data) > P(H0 | data) -- for example, whenever BF_10 < 1; in this case, the data are evidential for H0, not H1.
hellas flag behind i loled! go greece