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seen Mar 16 '12 at 13:38

Feb
26
comment Question on the notion of a $\sigma$-algebra generated by a function
I think when one talks about a function, then `by default' it takes values in $\mathbb R$, equipped with ${\cal B}(\mathbb R)$ (surely, one can equip $\mathbb R$ with a different $\sigma$-algebra... but this would always be specified). In general, in the case you discussed here I believe $f$ is referred to as a (measurable) mapping to $(E,{\cal E})$. This, of course, is a more general case.
Feb
26
comment Interpretation of sigma algebra
@Tim: I would recommend to follow a textbook instead of Wikipedia. In this case, you would have much less problems of consistency of definitions (most of the time, terminologies). A Probability Path by Resnick is a good one and not too mathy. If you are confident at math, then try Probability: Theory and Examples (math.duke.edu/~rtd/PTE/pte.html) by Rick Durrett.
Feb
26
answered Question on the notion of a $\sigma$-algebra generated by a function
Feb
26
comment Interpretation of sigma algebra
Tim: 3 is clear (but it is better to interpret the elements in the $\sigma$-albegra as events, as suggested by Qiaochu), while what you are wondering about in 1 and 2 are a little vague... BTW, which book are you using to study probabiilty?
Feb
22
answered How to understand marginal distribution
Feb
22
comment Measure from on product $\sigma$-algebra to on component $\sigma$-algebras
@Arturo This answers actually my question. Thanks.
Feb
22
comment Further questions about from product $\sigma$-algebra to component ones
@Arturo Right, (1) is a special case of (2). Thanks.
Feb
22
answered Notation question with Dirichlet processes
Feb
22
comment Measure from on product $\sigma$-algebra to on component $\sigma$-algebras
@Arturo: could you elaborate a little more on your last comment on the `measure invariant'? Thanks.
Feb
22
comment Further questions about from product $\sigma$-algebra to component ones
@Arturo: Thanks. I posted an answer. Please check and let me know if I made any mistakes.
Feb
22
answered Further questions about from product $\sigma$-algebra to component ones
Feb
22
comment How to understand marginal distribution
@Tim A little confusing here. Where do you want your $\mu_i$ to be defined? Anyway, $A_i\times \prod_{j\in I,j\neq i}X_j$ does not make sense. I think you mean $A_i\times \prod_{j\in I,j\neq i}S_j$, which is better written as $\pi_i^{-1}(A_i)$, where $\pi_i:\prod_{j\in I}S_j\to S_i$ is the projection mapping. Beside, $f$ should be involved somehow. Your second guess is on the right track. I let you finish the final answer. You are almost there.
Feb
22
comment Further questions about from product $\sigma$-algebra to component ones
Is your index set $I$ finite, countably many, or arbitrary?
Feb
22
comment Notation question with Dirichlet processes
@JasonMond: Sorry that I was misleading and confusing. I think your interpretation was correct. I was thinking of a slightly different type of processes... :P
Feb
22
comment Notation question with Dirichlet processes
@JasonMond: (1) I think the points $\theta_k$ can take values in even more abstract metric spaces (hence including the complex numbers and vectors), according your choice of $G_0$. (2) $G_0$ can well be a Bernoulli distribution. In this case, you can sample all the $\theta_k$'s as a process (infinitely many zeros and ones), or, I would add the $\beta_k$'s and see the obtained $G$ as a new (mixed, if $\beta_k$'s are random) Bernoulli distribution.
Feb
22
awarded  Commentator
Feb
22
comment Notation question with Dirichlet processes
Think of a discrete probability measure, then each point that has positive measure is an atom. For example, a Bernoulli distribution has atoms 0 and 1. Then, $G$ can be seen as a random discrete probability measure with atoms at $\theta_k$, with (random) mass $\beta_k$. To draw a sample according to $G$, it means to draw a point $\theta_k$ according to the weight $\beta_k$. (In general, DP processes can be formulated as point processes. But this might be too much for the time being.)
Feb
21
comment How to understand marginal distribution
@Tim: (1) In general, the random elements (here random vectors) are defined via measurable mappings, which are with respect to $\sigma$-finite measures. The questions here concern only the mapping and I don't see how the finiteness plays a crucial role. (2) All the measures you see in the first class of measure theory are $\sigma$-finite (unless explicitly said to be outer measure). Focus on these measures should be general enough.
Feb
21
comment How to understand marginal distribution
For your two definitions ($P_{X_i}$ and $P_{X_i}'$), relate them to $P$ and you can see they are equivalent. Furthermore, it should be clear that the finiteness does not play a role here. At last, I think by `any measure' you mean any $\sigma$-finite measure, and your argument would remain valid.
Feb
21
revised Independent exponentially distributed random variables
I corrected some latex typos.