Bayesian updating normal distribution

Bayesian updating normal distribution


If we have better than a subjective guess, for instance a worldwide sampling of data, we can estimate the mean and variance of this prior. The following example of estimating the API gravity or oil density may help to see the above formulas at work. I can recommend an explanation of the update process for a probability and for a normal distribution given by Jacobs , which is more complete than what I have given here and explains the derivation of the formulas. In case of no sample information in our local prospect area, all we can do is accept the prior distribution of observations as the "predictive distribution" and use it as input to Gaeapas. Note that for prospect appraisal, the data are almost exclusively mean values of a parameter, such as the mean porosity of a reservoir in a field, not individual porosities in sidewall plugs of pieces of cores. By a standard result on the factorization of probability density functions see also the introduction to Bayesian inference , we have that Therefore, the posterior distribution is a normal distribution with mean and variance. So the actual prior distribution is telling us how uncertain the mean is. For prospect appraisal this is usually a worldwide sampling as used in the Gaeapas appraisal program. Posterior distribution, the revised, or "updated" prior, based on the sample of new information 3. When a prior dataset can be roughly represented by a normal distribution, bayesian statistics show that sample information from the same process can be used to obtain a posterior normal distribution. Both the prior and the sample mean convey some information a signal about. By the way, an excellent source to see the math behind the updating procedure, including the treatment of the variance is given by Jacobs, In the limit, all weight is given to the information coming from the sample and no weight is given to the prior. The formulas involved are shown here without giving the derivation Jacob, , Winkler, We have yet to figure out what is. The latter is a weighted combination of the prior and the sample. Thus, the posterior distribution of is a normal distribution with mean and variance Note that the posterior mean is the weighted average of two signals: The greater the precision of a signal, the higher its weight is. The weight given to the sample mean increases with , while the weight given to the prior mean does not. The following data and calculation shows what happens in this example. The bayesian process of obtaining a posterior distribution of observations which can be used for sampling in a Monte Carlo procedure, uses the distribution of the mean of the observations. The mean of this distribution is the same as the posterior mean, but the variance of this mean is a weighted combination of process and posterior variance. This will be done in the next proof. If several new observations are made, the mean value of these is used and compared to the prior distribution. A prior distribution can be constructed by collecting data, or by "subjective experience" which can not be formally processed. The World-wide prior distributions we have assembled are distributions of observed values. In practice a compromise is made that is anyway much better than not using the worldwide background factual experience.

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Bayesian updating normal distribution

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31 - Normal prior conjugate to normal likelihood - proof 1




As a consequence, when the sample size becomes large, more and more weight is given to the sample mean. In the case of a world-wide prior see 2 , the new information may be observations that are local around the prospect to be evaluated, so-called "analogons". For prospect appraisal this is usually a worldwide sampling as used in the Gaeapas appraisal program. Both the prior and the sample mean convey some information a signal about. In that case it has a high likelihood, it "hits" the prior at the highest probability density. The greater the precision of a signal, the higher its weight is. The signals are combined linearly , but more weight is given to the signal that has higher precision smaller variance. However, preferably it is based on a collection of data, a sample. If sample information becomes available it may be that, for instance, a single new measurement "x" will not change our prior estimate when it is equal to the mean of the prior. Then I would use the regression to adjust this data to m depth as if the samples were at equal depth. It may be clear that, if there is no local information we bring to bear, the best guess is the prior distribution which then can not be updated. It combines in the simulation model the world-wide variation of a variable and the relevant local data to give the most realistic value and its uncertainty. This example is at m depth, so we have the mean of the process from the regression and the process variance as the square of the adjusted standard error of estimate, i.

Bayesian updating normal distribution


If we have better than a subjective guess, for instance a worldwide sampling of data, we can estimate the mean and variance of this prior. The following example of estimating the API gravity or oil density may help to see the above formulas at work. I can recommend an explanation of the update process for a probability and for a normal distribution given by Jacobs , which is more complete than what I have given here and explains the derivation of the formulas. In case of no sample information in our local prospect area, all we can do is accept the prior distribution of observations as the "predictive distribution" and use it as input to Gaeapas. Note that for prospect appraisal, the data are almost exclusively mean values of a parameter, such as the mean porosity of a reservoir in a field, not individual porosities in sidewall plugs of pieces of cores. By a standard result on the factorization of probability density functions see also the introduction to Bayesian inference , we have that Therefore, the posterior distribution is a normal distribution with mean and variance. So the actual prior distribution is telling us how uncertain the mean is. For prospect appraisal this is usually a worldwide sampling as used in the Gaeapas appraisal program. Posterior distribution, the revised, or "updated" prior, based on the sample of new information 3. When a prior dataset can be roughly represented by a normal distribution, bayesian statistics show that sample information from the same process can be used to obtain a posterior normal distribution. Both the prior and the sample mean convey some information a signal about. By the way, an excellent source to see the math behind the updating procedure, including the treatment of the variance is given by Jacobs, In the limit, all weight is given to the information coming from the sample and no weight is given to the prior. The formulas involved are shown here without giving the derivation Jacob, , Winkler, We have yet to figure out what is. The latter is a weighted combination of the prior and the sample. Thus, the posterior distribution of is a normal distribution with mean and variance Note that the posterior mean is the weighted average of two signals: The greater the precision of a signal, the higher its weight is. The weight given to the sample mean increases with , while the weight given to the prior mean does not. The following data and calculation shows what happens in this example. The bayesian process of obtaining a posterior distribution of observations which can be used for sampling in a Monte Carlo procedure, uses the distribution of the mean of the observations. The mean of this distribution is the same as the posterior mean, but the variance of this mean is a weighted combination of process and posterior variance. This will be done in the next proof. If several new observations are made, the mean value of these is used and compared to the prior distribution. A prior distribution can be constructed by collecting data, or by "subjective experience" which can not be formally processed. The World-wide prior distributions we have assembled are distributions of observed values. In practice a compromise is made that is anyway much better than not using the worldwide background factual experience.

Bayesian updating normal distribution


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2 thoughts on “Bayesian updating normal distribution

  1. So the actual prior distribution is telling us how uncertain the mean is. See also the page on priors.

  2. The probability density function pdf is: I can recommend an explanation of the update process for a probability and for a normal distribution given by Jacobs , which is more complete than what I have given here and explains the derivation of the formulas.

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