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I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you draw the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample meantotal of the samples, uncorrected sample variancetotal of the squared samples, and the number of samples. (Or any bijective map of these.)

Incidentally, this is why I like civilstat's answer because it highlights our desire to combine information. The three sufficient statistics above are even better than the two given in the question (or by civilstat's answer). Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you draw the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample mean, uncorrected sample variance, and number of samples.

Incidentally, this is why I like civilstat's answer. The three sufficient statistics above are even better. Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you draw the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are the total of the samples, total of the squared samples, and the number of samples. (Or any bijective map of these.)

Incidentally, I like civilstat's answer because it highlights our desire to combine information. The three sufficient statistics above are even better than the two given in the question (or by civilstat's answer). Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

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I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you drawndraw the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample mean, uncorrected sample variance, and number of samples.

Incidentally, this is why I like civilstat's answer. The three sufficient statistics above are even better. Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you drawn the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample mean, uncorrected sample variance, and number of samples.

Incidentally, this is why I like civilstat's answer. The three sufficient statistics above are even better. Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you draw the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample mean, uncorrected sample variance, and number of samples.

Incidentally, this is why I like civilstat's answer. The three sufficient statistics above are even better. Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.

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I want to add the Bayesian answer to this discussion. Just because your assumption is that the data is generated according to some normal with unknown mean and variance, that doesn't mean that you should summarize your data using a mean and a variance. This whole problem can be avoided if you drawn the model, which will have a posterior predictive that is a three parameter noncentral scaled student's T distribution. The three parameters are related to the sample mean, uncorrected sample variance, and number of samples.

Incidentally, this is why I like civilstat's answer. The three sufficient statistics above are even better. Two sets of these statistics can easily be combined, and they give the best posterior predictive given the assumption of normality.