microhaus
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The derivation relies on a result known as the matrix inversion lemma, or Woodbury matrix identity. From wikipedia: $$(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}$$ Identifying $... View answer 1 votes The class of methods devised to solve this generic problem are known in the optimisation literature as projected Newton methods. The following is extracted from Bertsekas and Gafni (1983), with some ... View answer Accepted answer 1 votes This answer assumes that you are interested in computing$\nabla_{\theta} \log Z$rather than$\nabla_{x} \log Z$. Without foreclosing the possibility that there may exist other results relevant to ... View answer Accepted answer 1 votes Only in response to: I'm looking for a reference that talks about the relationship between the variance of the gradient and convergence of SGD. Whilst the Robbins-Monro stochastic approximation ... View answer Accepted answer 1 votes I think it would be helpful in this instance to draw a distinction between: 1. How the hyperparametrisation encodes prior beliefs about the topic vectors$\phi_k$sampled from the Dirichlet prior$p(\...

Some references. To supplement the comments of @Dave Harris, here are a cluster of references you might consider using to start further formalising what you've done already. Depending on your ...

In response to: So how do I find $E(x^2_i)$? Assuming there are no issues with the steps in the derivation up to that point, you can use the following standard result relating expectation, variance ...

From the context you have provided, my reading is that $x^{(i)}_j$ is the $j$-th element of the $i$-th input vector $\mathbf{x}^{(i)}$, where there are $i = 1,..., m$ training instances. Addressing ...

Extended comment. I am somewhat confused by what it is you are seeking an answer for, and also by your workings. On the workings: Assuming that by $f(x^n; \theta)$ you refer to the likelihood function ...

The statement the authors intended is (A) for the following reasons. Addressing the confusion. Much of your reluctance to consider $(A)$ as the correct statement amounted to insufficient attentiveness ...

You understanding is completely correct. Starting at step 1, you initialise uniform weights $w_i = 1 / N$. Beginning at step 2, in say iteration $m = 1$, you sample a bootstrap dataset $\mathcal{B}... View answer Accepted answer 1 votes The following lecture video, "L10 - Gaussian graphical models and Ising models", here of the course CMU Probabilistic Graphical Models Spring 2019 has a small section towards the end about ... View answer Accepted answer 1 votes Typo. There is a typo in the 2006 version of Bishop's PRML, which has been corrected in later versions. Instead, equation (9.39) on p443 should read "the expectation of the indicator variable$z_{...

The question you raised is down to confusion concerning whether or not the $\beta_{k, \space :}$ is drawn from an identically parametrised Dirichlet distribution, or whether each $\beta_{k, \space :}$ ...

Fix $z \in \mathbb{R}$ to be some value. Now for illustrative purposes, define corresponding i.i.d. random variables $Y_i = g(Z_i) = 1\{Z_i \leq z\}$. The following prompts should hopefully assist in ...

\begin{align} p(t \vert \mathbf{x}, \mathcal{D}) &= \sum^L_{i=1}p(t, \mathcal{M}_i \vert \mathbf{x}, \mathcal{D}) \tag{marginalisation} \\ &= \sum^L_{i=1} p(t \vert \mathbf{x}, \mathcal{M_i}, \...

Solution. Using the results in here linked by StubbornAtom. The loss function you have specified can be interpreted as a generalisation of squared error loss, known as weighted squared error loss. ...

Here is my attempt. 1.Show that $f(x; \varphi)$ is in the one-parameter exponential family. Assume that $\varphi > 0$. Rewriting $f(x; \varphi)$, we have that \begin{align*} f(x; \varphi) &= \...

It depends on the nature of the assumptions on your $n$ random variables $Y_1, ..., Y_n$, which in turn will influence the kind of parametric statistical model that is specified. Here is a coin ...

In response to I did not fully understand why we are multiplying the intermediate conditionals i.e.: $...Pr(Burglar|Storm=T)xPr(Cat|Storm=T)...$ and Is there a logical explanation for the ...

In response to question and also following comment: Sorry, but could I ask a small follow-up question? If θj = 5 then should the jth element in θ also be 5? E.g - If on the left hand side θj= θ3 = 5 ...

The use of the index $l$ is just a 'dummy indexing variable' - it is used to avoid confusion with the fact that the indexing variable $k$ has already been used in the numerator. You generally tend to ...