Mathematics

Questions tagged [expectation-maximization]

13 questions
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$30$ Sided Dice Problem. Interview Preparation.

I came through this problem three years ago when I prepared for interview. Today when reviewing it, I still couldn't think out a mathematical way to solve it. Any help would be appreciated. $4$ players roll a fair $30$ -sided dice. Each of them…
ggkkll
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3
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Geometry of payoffs in a 3+ player zero-sum game

In a $2$ player zero-sum game in which each player may mix between $2$ pure actions, each player's action space is an interval in $\mathbb{R}^2$ ( $1$ dimension of the ambient space is constrained by the axiom that probabilities sum to $1$ , and the…
user10478
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2
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The upper bound for the max of absolute values of Gaussian distribution

Prove that $E[\max_{1\leq i\leq n}\vert X_i\vert]\leq C\sqrt{\log(2n)}\sigma$ for some constant $C \in R$ , where $X_1, ..., X_n\sim \mathcal{N}(0,\sigma)$ are i.i.d. Hint: Use Jensen's inequality: $f(E[Y])\leq E[f(Y)]$ for any convex function $f$ .…
Nick Y
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Convergence of the Expectation-Maximization algorithm

Studying the Expectation-Maximization algorithm, I noticed that I couldn't find any proof that the parameters actually converge, nor that the limit is a local extremum of the likelihood (or even just a point where the gradient vanishes). The proofs…
user25640
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Expectation value given a convex order

I want to know if the following proof is correct. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \begin{equation} \mathbb{E}[u(N)]\leq\mathbb{E}[u(M)] \end{equation} Then…
Don P.
  • 438
1
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1 answer

Binary choice with mixture of normal errors estimation

My problem is simplified as follows: I'm estimating a binary choice probit model where the error term follows a mixture of two normal distributions. This is: $y_i = \mathbf{1} \left( \beta_0 + \beta_1 x_i + \epsilon_i > 0 \right)$ Where $\epsilon_i…
Mark F
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Discretising variational problems through linear programming

Suppose we have the following variational problem. Given $\mu>1$ and the function $g \in C[0,1]$ , $$ \begin{array}{ll} \underset{f \in C[0,1]}{\text{maximize}} & \overbrace{\displaystyle\int_0^1 x f(x) \, {\rm d} x}^{=: I(f)} \\ \text{subject to} &…
xyz
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Maximizing expected gains in game of box openings

Question Coins are located in boxes, but the content of each box is unknown. A box has a probability $p$ of containing a coin, but a probability $1-p$ of forcing its opener to quit the game (and lose all coins). What number of boxes $k$ would need…
HappyDay
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In the limit, what probability of tails maximizes the expected number of shorter runs of heads followed by a longer run of tails.

We flip a coin $n$ times. The probability of tails is $p$ . Let $X$ be the number of occurrences of a run of heads followed by a longer run of tails. We can also agree that if the sequence begins with a run of tails that is also an occurrence…
ploosu2
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maximum of random matrix

Let $(M_{k,n})_{\substack{0\leqslant k\leqslant K\\ 0\leqslant n \leqslant N}}$ a matrix of nonnegative random variables and let $C$ be a constant. Assume that for all $0\leqslant n\leqslant N$ , $\mathbb{E}\big[\max_{0\leqslant k\leqslant K}…
20Xblog8x12
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0
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Expectation of maximum of constant minus normal distribution and zero

All day, I've been busy with solving the following expectation: $E[Max(c-Y,0)]$ Where $c$ is a constant, $Y$ is normally distributed with mean $μ$ and standard deviation $σμ^β$ . I tried using the fact that for a standard normal random variable $Z$…
TimonSeldenrijk
  • 111
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Expected payoff in a dice game

I was solving questions for finding expectation values, I enocuntered a question which is bugging me, the question says, You are given a fair $6$ sided die and you roll it. You can either choose to keep your roll and receive the observed value in…
Shriom707
  • 333
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Trying to understand an application of expectation-Maximization algorithm

In the paper Johannes Kopf, Ariel Shamir, Pieter Peers - Content Adaptive Image Downscaling , the authors apply the Expectation-Maximization algorithm to solve an image downscaling problem. I've never done Bayesian statistics, so I find some parts a…
Babis
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