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Bayesian updating normal prior

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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. The signals are combined linearly , but more weight is given to the signal that has higher precision smaller variance. This has the disadvantage that it does not account for any uncertainty in the value of the parameter, and hence will underestimate the variance of the predictive distribution. 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: Inference over exclusive and exhaustive possibilities[ edit ] If evidence is simultaneously used to update belief over a set of exclusive and exhaustive propositions, Bayesian inference may be thought of as acting on this belief distribution as a whole. In the limit, all weight is given to the information coming from the sample and no weight is given to the prior. In Bayesian statistics, however, the posterior predictive distribution can always be determined exactly—or at least, to an arbitrary level of precision, when numerical methods are used. Starting at different points yields different flows over time. As a consequence, when the sample size becomes large, more and more weight is given to the sample mean. This greatly simplifies the analysis, as it otherwise considers an infinite-dimensional space space of all functions, space of all distributions. In fact, if the prior distribution is a conjugate prior , and hence the prior and posterior distributions come from the same family, it can easily be seen that both prior and posterior predictive distributions also come from the same family of compound distributions. In all cases below, the data is assumed to consist of n points x. Interpretations[ edit ] Analogy with eigenfunctions[ edit ] Conjugate priors are analogous to eigenfunctions in operator theory , in that they are distributions on which the "conditioning operator" acts in a well-understood way, thinking of the process of changing from the prior to the posterior as an operator. For example, confidence intervals and prediction intervals in frequentist statistics when constructed from a normal distribution with unknown mean and variance are constructed using a Student's t-distribution. The weight given to the sample mean increases with , while the weight given to the prior mean does not.

Bayesian updating normal prior


As a consequence, when the sample size becomes large, more and more weight is given to the sample mean. In the limit, all weight is given to the information coming from the sample and no weight is given to the prior. Both the prior and the sample mean convey some information a signal about. For related approaches, see Recursive Bayesian estimation and Data assimilation. This can help both in providing an intuition behind the often messy update equations, as well as to help choose reasonable hyperparameters for a prior. This is again analogous with the dynamical system defined by a linear operator, but note that since different samples lead to different inference, this is not simply dependent on time, but rather on data over time. In both eigenfunctions and conjugate priors, there is a finite-dimensional space which is preserved by the operator: For example, confidence intervals and prediction intervals in frequentist statistics when constructed from a normal distribution with unknown mean and variance are constructed using a Student's t-distribution. Just as one can easily analyze how a linear combination of eigenfunctions evolves under application of an operator because, with respect to these functions, the operator is diagonalized , one can easily analyze how a convex combination of conjugate priors evolves under conditioning; this is called using a hyperprior , and corresponds to using a mixture density of conjugate priors, rather than a single conjugate prior. Interpretations[ edit ] Analogy with eigenfunctions[ edit ] Conjugate priors are analogous to eigenfunctions in operator theory , in that they are distributions on which the "conditioning operator" acts in a well-understood way, thinking of the process of changing from the prior to the posterior as an operator. In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. In all cases below, the data is assumed to consist of n points x. However, the processes are only analogous, not identical: Only this way is the entire posterior distribution of the parameter s used. Note that both types of predictive distributions have the form of a compound probability distribution as does the marginal likelihood. The prior predictive distribution The prior predictive distribution is where. 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. Dynamical system[ edit ] One can think of conditioning on conjugate priors as defining a kind of discrete time dynamical system: This will be done in the next proof. This greatly simplifies the analysis, as it otherwise considers an infinite-dimensional space space of all functions, space of all distributions. We have yet to figure out what is. The only difference is that the posterior predictive distribution uses the updated values of the hyperparameters applying the Bayesian update rules given in the conjugate prior article , while the prior predictive distribution uses the values of the hyperparameters that appear in the prior distribution. The greater the precision of a signal, the higher its weight is. In some instances, frequentist statistics can work around this problem. In fact, if the prior distribution is a conjugate prior , and hence the prior and posterior distributions come from the same family, it can easily be seen that both prior and posterior predictive distributions also come from the same family of compound distributions. Inference over exclusive and exhaustive possibilities[ edit ] If evidence is simultaneously used to update belief over a set of exclusive and exhaustive propositions, Bayesian inference may be thought of as acting on this belief distribution as a whole. The signals are combined linearly , but more weight is given to the signal that has higher precision smaller variance.

Bayesian updating normal prior


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