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In Bayesian statisticsthe posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. Given a set of N i. The prior predictive distributionin a Bayesian context, is the distribution of a data point marginalized over its prior distribution. This is similar to the posterior predictive distribution except that the marginalization or equivalently, expectation is taken with respect to the prior distribution instead of the posterior distribution.

This is easy to see. Hence, the posterior predictive distribution follows the same distribution H as the prior predictive distribution, but with the posterior values of the hyperparameters substituted for the prior ones. In some cases the appropriate compound distribution is defined using a different parameterization than the one that would be most natural for the predictive distributions in the current problem at hand.

Often this results because the prior distribution used to define the compound distribution is different from the one used in the current problem. For example, as indicated above, the Student's t-distribution was defined in terms of a scaled-inverse-chi-squared distribution placed on the variance.

However, it is more common to use an inverse gamma distribution as the conjugate prior in this situation. The two are in fact equivalent except for parameterization; hence, the Student's t-distribution can still be used for either predictive distribution, but the hyperparameters must be reparameterized before being plugged in.

Most, but not all, common families of distributions belong to the exponential family of distributions. Exponential families have a large number of useful properties. One of which is that all members have conjugate prior distributions — whereas very few other distributions have conjugate priors. Another useful property is that the probability density function of the compound distribution corresponding to the prior predictive distribution of an exponential family distribution marginalized over its conjugate prior distribution can be determined analytically.

Hence the result of the integration will be the reciprocal of the normalizing function. The reason the integral is tractable is that it involves computing the normalization constant of a density defined by the product of a prior distribution and a likelihood. When the two are conjugatethe product is a posterior distributionand by assumption, the normalization constant of this distribution is known.

The beta-binomial distribution is a good example of how this process works. Despite the analytical tractability of such distributions, they are in themselves usually not members of the exponential family.Prior probability is the probability of an event before we see the data.

In Bayesian Inferencethe prior is our guess about the probability based on what we know now, before new data becomes available. Conjugate prior just can not be understood without knowing Bayesian inference. For some likelihood functions, if you choose a certain prior, the posterior ends up being in the same distribution as the prior.

Such a prior then is called a Conjugate Prior. It is a lways best understood through examples. Below is the code to calculate the posterior of the binomial likelihood. A question to you: Is there anything that concerns you in the code block above?

There are two things that make the posterior calculation expensive. Why do we have to calculate the posterior for thousands of thetas? Because you are normalizing the posterior line Even if you choose not to normalize the posterior, the end goal is to find the maximum of the posteriors Maximum a posteriori.

Second, if there is no closed-form formula of the posterior distribution, we have to find the maximum by numerical optimization, such as gradient descent or newtons method. Furthermore, if your prior distribution has a closed-form form expression, you already know what the maximum posterior is going to be. In the example above, the beta distribution is a conjugate prior to the binomial likelihood.

What does this mean? It means during the modeling phase, we already know the posterior will also be a beta distribution. This is very convenient! Proof in the next section. As you saw, the computations in Bayesian Inference can be heavy or sometimes even intractable.

However, if we could use the closed-form formula of the conjugate prior, the computation becomes very light. When we use the Beta distribution as a prior, a posterior of binomial likelihood will also follow the beta distribution.It is a multivariate generalization of the beta distribution[1] hence its alternative name of multivariate beta distribution MBD. The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process. The normalizing constant is the multivariate beta functionwhich can be expressed in terms of the gamma function :.

These can be viewed as the probabilities of a K -way categorical event. Another way to express this is that the domain of the Dirichlet distribution is itself a set of probability distributionsspecifically the set of K -dimensional discrete distributions.

The symmetric case might be useful, for example, when a Dirichlet prior over components is called for, but there is no prior knowledge favoring one component over another. This particular distribution is known as the flat Dirichlet distribution.

Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i. Values of the concentration parameter below 1 prefer sparse distributions, i. The concentration parameter in this case is larger by a factor of K than the concentration parameter for a symmetric Dirichlet distribution described above. This construction ties in with concept of a base measure when discussing Dirichlet processes and is often used in the topic modelling literature.

Then [4] [5]. The matrix so defined is singular. More generally, moments of Dirichlet-distributed random variables can be expressed as [6].

The mode of the distribution is [7] the vector x 1The marginal distributions are beta distributions : [8]. The Dirichlet distribution is the conjugate prior distribution of the categorical distribution a generic discrete probability distribution with a given number of possible outcomes and multinomial distribution the distribution over observed counts of each possible category in a set of categorically distributed observations. This means that if a data point has either a categorical or multinomial distribution, and the prior distribution of the distribution's parameter the vector of probabilities that generates the data point is distributed as a Dirichlet, then the posterior distribution of the parameter is also a Dirichlet.

Intuitively, in such a case, starting from what we know about the parameter prior to observing the data point, we then can update our knowledge based on the data point and end up with a new distribution of the same form as the old one. This means that we can successively update our knowledge of a parameter by incorporating new observations one at a time, without running into mathematical difficulties.For example, the Gaussian family is conjugate to itself or self-conjugate with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian.

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This means that the Gaussian distribution is a conjugate prior for the likelihood that is also Gaussian.

The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory. Let the likelihood function be considered fixed; the likelihood function is usually well-determined from a statement of the data-generating process [ example needed ].

For certain choices of the prior, the posterior has the same algebraic form as the prior generally with different parameter values. Such a choice is a conjugate prior. A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise numerical integration may be necessary. Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution.

All members of the exponential family have conjugate priors. The form of the conjugate prior can generally be determined by inspection of the probability density or probability mass function of a distribution. This random variable will follow the binomial distributionwith a probability mass function of the form.

It is a typical characteristic of conjugate priors that the dimensionality of the hyperparameters is one greater than that of the parameters of the original distribution. If all parameters are scalar values, then this means that there will be one more hyperparameter than parameter; but this also applies to vector-valued and matrix-valued parameters.

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See the general article on the exponential familyand consider also the Wishart distributionconjugate prior of the covariance matrix of a multivariate normal distributionfor an example where a large dimensionality is involved. This posterior distribution could then be used as the prior for more samples, with the hyperparameters simply adding each extra piece of information as it comes.

It is often useful to think of the hyperparameters of a conjugate prior distribution as corresponding to having observed a certain number of pseudo-observations with properties specified by the parameters. In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. 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.

Conjugate priors are analogous to eigenfunctions in operator theoryin 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 both eigenfunctions and conjugate priors, there is a finite-dimensional space which is preserved by the operator: the output is of the same form in the same space as the input. This greatly simplifies the analysis, as it otherwise considers an infinite-dimensional space space of all functions, space of all distributions.

However, the processes are only analogous, not identical: conditioning is not linear, as the space of distributions is not closed under linear combinationonly convex combinationand the posterior is only of the same form as the prior, not a scalar multiple. 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 diagonalizedone can easily analyze how a convex combination of conjugate priors evolves under conditioning; this is called using a hyperpriorand corresponds to using a mixture density of conjugate priors, rather than a single conjugate prior.

One can think of conditioning on conjugate priors as defining a kind of discrete time dynamical system : from a given set of hyperparameters, incoming data updates these hyperparameters, so one can see the change in hyperparameters as a kind of "time evolution" of the system, corresponding to "learning". Starting at different points yields different flows over time. 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.

For related approaches, see Recursive Bayesian estimation and Data assimilation. Suppose a rental car service operates in your city. Drivers can drop off and pick up cars anywhere inside the city limits. You can find and rent cars using an app. Suppose you wish to find the probability that you can find a rental car within a short distance of your home address at any given time of day.

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But the data could also have come from another Poisson distribution, e. In fact there is an infinite number of poisson distributions that could have generated the observed data and with relatively few data points we should be quite uncertain about which exact poisson distribution generated this data.Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

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It only takes a minute to sign up. I've been looking for simple code that can model ad clicks per day. Notionally, gamma-poisson would be a good conjugate prior. However, I'm finding that for slightly large daily click rate values, the denominator, n-1! And the resulting plot:. As you can see from code, my prior belief was that the rate was 2 clicks per day. In truth this is simulated data and the actual rate is 4.

The plot does slowly converge, however, the peak shrinks quite a bit and isn't necessary tightening the variance.

I've used similar code for a Beta-Binomial conjugate prior and the results were night and day different. In the beta case, the peaks increased and became tighter with more data. In the gamma case, the peaks reduced and ultimately the code crashed after 40 of 50 iterations because the denominator exploded.

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## Conjugate prior

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### Posterior predictive distribution

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