Sự khác biệt giữa phân phối dự báo sau và sau là gì?

Tôi hiểu Posterior là gì, nhưng tôi không chắc cái sau có nghĩa là gì?

Làm thế nào là 2 khác nhau?

Kevin P Murphy đã chỉ ra trong sách giáo khoa của mình, Machine Learning: một quan điểm xác suất , rằng đó là "một trạng thái niềm tin nội bộ". Điều đó thực sự có ý nghĩa gì? Tôi đã có ấn tượng rằng một Ưu tiên đại diện cho niềm tin hoặc thành kiến nội bộ của bạn, tôi đang sai ở đâu?

posterior definition

— QUẢNG CÁO
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Câu trả lời:

$\theta$

p (θ | x) = c \times p (x | θ) p (θ)

$p(\theta|x)=c\times p(x|\theta)p(\theta)$

c

$c$

$\theta$

p (x^{*} | x) = \int_{Θ} c \times p (x^{*}, θ | x) d θ = \int_{Θ} c \times p (x^{*} | θ) p (θ | x) d θ

$p(x^*|x)=\int_\Theta c\times p(x^*,\theta|x)d\theta=\int_\Theta c\times p(x^*|\theta)p(\theta|x)d\theta$

where $x^*$ is a new unobserved random variable and is independent of $x$ .

I won't dwell on the posterior distribution explanation since you say you understand it but the posterior distribution "is the distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained" (Wikipedia). So basically its the distribution that explains your unknown, random, parameter.

On the other hand, the posterior predictive distribution has a completely different meaning in that it is the distribution for future predicted data based on the data you have already seen. So the posterior predictive distribution is basically used to predict new data values.

If it helps, is an example graph of a posterior distribution and a posterior predictive distribution:

enter image description here

— Jinhua Wang
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That posterior predictive distribution graph needs new axis labels and a caption or something. I get the idea because I know what a posterior predictive distribution is, but someone who's just figuring it out could get seriously confused.

— Cyan

Thanks @BabakP could you also point me to what what distribution you used to plot the pmf of theta, and P(x*|theta)

— A.D

...cause I would like to work out the full example.

— A.D

I just pretended that my posterior was a Beta(3,2). I did not actually work out anything. But of course, if you want an example, assume the likelihood is a Binomial(n,p) and the prior on p is a Beta(a,b) then you should be able to obtain that the posterior is once again a beta distribution.

As well, that posterior predictive is not an easy one to derive. I just grabbed a graph from some Gaussian Process code I wrote for a GP posterior predictive. And with that said, that posterior and that posterior predictive plot above does not actually correspond to the posterior shown, they are both arbitrary.

The predictive distribution is usually used when you have learned a posterior distribution for the parameter of some sort of predictive model. For example in Bayesian linear regression, you learn a posterior distribution over the w parameter of the model y=wX given some observed data X.
Then when a new unseen data point x* comes in, you want to find the distribution over possible predictions y* given the posterior distribution for w that you just learned. This distribution over possible y*'s given the posterior for w is the prediction distribution.

— user1893354
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They refer to distributions of two different things.

The posterior distribution refers to the distribution of the parameter, while the predictive posterior distribution (PPD) refers to the distribution of future observations of data.

— SPMQET
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