Imprecise probabilistic models for inference in exponential families - - PowerPoint PPT Presentation

Sketch of the context The specifics Summary

Imprecise probabilistic models for inference in exponential families

Erik Quaeghebeur Gert de Cooman

SYSTeMS reseach group Ghent University

2 June 2006

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Sketch of the context

There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process.

Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s).

Inferences are typically expressed using

probabilities of events, or previsions of gambles.

The sample size is possibly small, so instead use

lower & upper probabilities of events, or lower & upper previsions of gambles.

A classical inference model structure is used:

Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For one sample x, Efψ(x) = a(x) exp (ψ, τ(x) − b(ψ)). For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Exponential family sampling models

Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form:

For a sequence x of m samples, Efψ(x) = a(x) expm (ψ, τ(x) − b(ψ)).

Other concepts:

SEfx(ψ) = Efψ(x), Sufficient statistic (m, τ(x)), and The likelihood function LEfm,τ(x)(ψ) = expm (ψ, τ(x) − b(ψ)).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The foundations: conjugate & predictive distributions

Basis of the prior & posterior models: conjugate families of distributions. Typical conjugate family form: CEfn,y(ψ) = c(n, y) expn (ψ, y − b(ψ)). Updating is done using Bayes’s rule: CEfn0,y0LEfm,τ(x) ∝ CEfn0+m, n0y0+mτ(x)

n0+m

. CEfn,y is the basis for the parametric inference models. The predictive family of distributions is derived from the conjugate family: PEfn,y(x) =

• Ψ

CEfn,ySEfx = c(n, y)a(x) c(n + m, ny+mτ(x)

n+m

) . PEfn,y is the basis for the parametric inference models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models I

The inference models are linear previsions defined using a distribution:

The conjugate distribution is used for the parametric model, PC(f | n, y) =

• Ψ

f CEfn,y, the predictive distribution is used for the predictive model, PP(g | n, y) =

• X ∗

m

gCEfn,y.

The prevision of particular gambles is easy to calculate:

Considering that ∇ψb(ψ) =

• X ∗

m τEfψ, it is a nice result for

the parametric model that PC(∇b | n, y) = y. An analogous result holds for the predictive model: PP(τ | n, y) = y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models I

The inference models are linear previsions defined using a distribution:

The conjugate distribution is used for the parametric model, PC(f | n, y) =

• Ψ

f CEfn,y, the predictive distribution is used for the predictive model, PP(g | n, y) =

• X ∗

m

gCEfn,y.

The prevision of particular gambles is easy to calculate:

Considering that ∇ψb(ψ) =

• X ∗

m τEfψ, it is a nice result for

the parametric model that PC(∇b | n, y) = y. An analogous result holds for the predictive model: PP(τ | n, y) = y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models I

The inference models are linear previsions defined using a distribution:

The conjugate distribution is used for the parametric model, PC(f | n, y) =

• Ψ

f CEfn,y, the predictive distribution is used for the predictive model, PP(g | n, y) =

• X ∗

m

gCEfn,y.

The prevision of particular gambles is easy to calculate:

Considering that ∇ψb(ψ) =

• X ∗

m τEfψ, it is a nice result for

the parametric model that PC(∇b | n, y) = y. An analogous result holds for the predictive model: PP(τ | n, y) = y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models II

Problem: how to choose n0 and y0? Effect on the parameters of updating with a sequence of m samples x: (nm, ym) = (n0 + m, n0y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models II

Problem: how to choose n0 and y0? Effect on the parameters of updating with a sequence of m samples x: (nm, ym) = (n0 + m, n0y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘precise’ inference models II

Problem: how to choose n0 and y0? Effect on the parameters of updating with a sequence of m samples x: (nm, ym) = (n0 + m, n0y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘precise’ inference models: normal

z1 z2 T = {z | z2 = z12} x1 x2 τ(x1) τ(x2) τ(x) y0 y1 y2

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘precise’ inference models: Bernoulli

z1 z2 1 1 T = {(0, 0), (1, 0), (0, 1)} x1 x2 τ(x) y0 y1 y2

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘precise’ inference models: von Mises

z1 z2 T = S2 = {z | z = 1} x1 x2 τ(x) y0 y1 y2

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models I

The inference models are lower previsions defined using sets of distributions:

Sets of conjugate distribution for the parametric model, PC(f | n, Y) = inf

y∈Y PC(f | n, y),

sets of predictive distributions for the predictive model, PP(g | n, Y) = inf

y∈Y PP(g | n, y).

The lower prevision of some gambles is again easy to calculate:

The nice result for the parametric model is PC(∇b | n, Y) = inf

y∈Y y.

The analogous result for the predictive model: PP(τ | n, Y) = inf

y∈Y y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models I

The inference models are lower previsions defined using sets of distributions:

Sets of conjugate distribution for the parametric model, PC(f | n, Y) = inf

y∈Y PC(f | n, y),

sets of predictive distributions for the predictive model, PP(g | n, Y) = inf

y∈Y PP(g | n, y).

The lower prevision of some gambles is again easy to calculate:

The nice result for the parametric model is PC(∇b | n, Y) = inf

y∈Y y.

The analogous result for the predictive model: PP(τ | n, Y) = inf

y∈Y y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models I

The inference models are lower previsions defined using sets of distributions:

Sets of conjugate distribution for the parametric model, PC(f | n, Y) = inf

y∈Y PC(f | n, y),

sets of predictive distributions for the predictive model, PP(g | n, Y) = inf

y∈Y PP(g | n, y).

The lower prevision of some gambles is again easy to calculate:

The nice result for the parametric model is PC(∇b | n, Y) = inf

y∈Y y.

The analogous result for the predictive model: PP(τ | n, Y) = inf

y∈Y y.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models II

Problem: how to choose n0 and Y0? Effect on the parameters of updating with a sequence of m samples x: (nm, Ym) = (n0 + m, n0Y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models II

Problem: how to choose n0 and Y0? Effect on the parameters of updating with a sequence of m samples x: (nm, Ym) = (n0 + m, n0Y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

The ‘imprecise’ inference models II

Problem: how to choose n0 and Y0? Effect on the parameters of updating with a sequence of m samples x: (nm, Ym) = (n0 + m, n0Y0 + mτ(x) n0 + m ).

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘imprecise’ inference models: normal

x1 x2 z1 z2 Y0 Y1 Y2 T = {z | z2 = z12} τ(x1) τ(x2) τ(x)

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘imprecise’ inference models: Bernoulli

T = {(0, 0), (1, 0), (0, 1)} z1 z2 Y0 Y1 Y2 1 1 x1 x2 τ(x)

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Updating the ‘imprecise’ inference models: von Mises

z1 z2 Y0 Y1 Y2 T = S2 = {z | z = 1} x1 x2 τ(x)

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Other uses of the inference models

Calculate lower and upper cumulative distribution functions. Combine inference models for multiple stochastic processes:

Combining independent marginals, or combining marginal and conditional models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Other uses of the inference models

Calculate lower and upper cumulative distribution functions. Combine inference models for multiple stochastic processes:

Combining independent marginals, or combining marginal and conditional models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Other uses of the inference models

Calculate lower and upper cumulative distribution functions. Combine inference models for multiple stochastic processes:

Combining independent marginals, or combining marginal and conditional models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary Exponential family sampling models The fundaments: conjugate & predictive distributions The parametric & predictive inference models

Other uses of the inference models

Calculate lower and upper cumulative distribution functions. Combine inference models for multiple stochastic processes:

Combining independent marginals, or combining marginal and conditional models.

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Summary

For all exponential family sampling models, imprecise probabilistic parametric and predictive inference models can be defined. Updating the models with sample data consist of a straightforward modification of the model parameters. Some inferences can be obtained very easily. Open questions:

What other inferences can be calculated? How do the inferences depend on n? How do these models compare with other models?

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Summary

For all exponential family sampling models, imprecise probabilistic parametric and predictive inference models can be defined. Updating the models with sample data consist of a straightforward modification of the model parameters. Some inferences can be obtained very easily. Open questions:

What other inferences can be calculated? How do the inferences depend on n? How do these models compare with other models?

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Summary

For all exponential family sampling models, imprecise probabilistic parametric and predictive inference models can be defined. Updating the models with sample data consist of a straightforward modification of the model parameters. Some inferences can be obtained very easily. Open questions:

What other inferences can be calculated? How do the inferences depend on n? How do these models compare with other models?

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Summary

For all exponential family sampling models, imprecise probabilistic parametric and predictive inference models can be defined. Updating the models with sample data consist of a straightforward modification of the model parameters. Some inferences can be obtained very easily. Open questions:

What other inferences can be calculated? How do the inferences depend on n? How do these models compare with other models?

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families

Sketch of the context The specifics Summary

Summary

For all exponential family sampling models, imprecise probabilistic parametric and predictive inference models can be defined. Updating the models with sample data consist of a straightforward modification of the model parameters. Some inferences can be obtained very easily. Open questions:

What other inferences can be calculated? How do the inferences depend on n? How do these models compare with other models?

Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families