Timeline for What mathematically enables one to Bayesian update multiple times?
Current License: CC BY-SA 4.0
13 events
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| Dec 30, 2019 at 17:38 | vote | accept | Noah Stebbins | ||
| May 7, 2019 at 5:47 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 7, 2019 at 0:42 | comment | added | dnqxt | Where do you see independence in that equality? | |
| May 7, 2019 at 0:25 | comment | added | Noah Stebbins | Hey @dnqxt, I'm not quite sure I follow. If the $x_i$-s are not assumed conditionally independent, then how does $p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n}, \theta) = p(x_{1:n+1}|\theta)$ in the second approach? | |
| May 7, 2019 at 0:09 | comment | added | dnqxt | That's a good question, but the answer is no, $x_i$-s are not assumed conditionally independent given the initial $\theta$. $\theta$ changes at each stage if one adopts the second (step by step or online) approach above. That's why it's necessary to include all $x_i$-s in the "batch" likelihood in the first approach according to the product rule. As for conditional independence of $x_i$-s given $\theta$ that indeed shows up in a popular Naive Bayes model, but that's another topic. | |
| May 6, 2019 at 23:48 | comment | added | Noah Stebbins | Hi @dnqxt, quick follow-up question: how did you go from $p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n}, \theta)$ to $p(x_{1:n+1}|\theta)$? Did you assume conditional independence between $x_{1:n}$ and $x_{n+1}$ ergo $p(x_{n+1}|x_{1:n}, \theta) = p(x_{n+1}|\theta)$ followed by using conditional independence between $x_{1:n}$ and $x_{n+1}$ again, ergo $p(x_{1:n+1}|\theta) = p(x_{1:n}|\theta)p(x_{n+1}|\theta)$? Thus, is conditional independence between $x_{1:n}$ and $x_{n+1}$ necessary to yield the same posterior distribution? | |
| May 6, 2019 at 2:28 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 21:46 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 21:38 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 21:04 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 20:36 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 20:28 | history | edited | dnqxt | CC BY-SA 4.0 |
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| May 5, 2019 at 20:15 | history | answered | dnqxt | CC BY-SA 4.0 |