## Solved – Softmax with log-likelihood cost

I am working on my understanding of neural networks using Michael Nielsen's "Neural networks and deep learning." Now in the third chapter, I am trying to develop an intuition of how softmax works together with a log-likelihood cost function. http://neuralnetworksanddeeplearning.com/chap3.html Nielsen defines the log-likelihood cost associated with a training input (eq. 80) as \$\$C equiv … Read more

## Solved – MLE of \$f(xverttheta)=1/theta\$, \$x_1 , cdots , x_n sim U(0,theta) ;;, theta>0\$,

Original question \$x_1 , cdots , x_n\$ are independent random variables, identically distributed as a uniform distribution over \$(0,theta)\$. \$\$ f(x vert theta) = frac{1}{theta}, ; 0<x<theta, ;; theta >0 \$\$ What's the Maximum Likelihood Estimator for \$theta\$. Comment on strict inequalities (from olooney, edited slightly) The MLE does not exist if we use strict … Read more

## Solved – Deriving likelihood function of binomial distribution, confusion over exponents

This question focuses on a specific aspect of this one: How to derive the likelihood function for binomial distribution for parameter estimation? In my own derivation, I start with: \$\$f(xmid p) = mC_x~p^x(1-p)^{m-x}\$\$ Ignoring \$mC_x\$, the likelihood function is then given by: \$\$L(p) = prod_{i=1}^np^{x_i}(1-p)^{m-x_i} = p^{sum_1^n x_i}(1-p)^{sum_1^n m-x_i} = p^{x}(1-p)^{nm-x}\$\$ However, in the question … Read more

## Solved – Validity of maximising log-likelihood for maximum likelihood estimation

For reasons owing to mathematical convenience, when finding MLEs (maximum likelihood estimates), it is often the log-likelihood function—as opposed to the standard likelihood function—which is maximised. From what I've gathered, this approach is deemed valid as a result of the monotonically increasing nature of the (natural) logarithm function. My understanding of a monotonically increasing function … Read more

## Solved – How to rigorously define the likelihood

The likelihood could be defined by several ways, for instance : the function \$L\$ from \$Thetatimes{cal X}\$ which maps \$(theta,x)\$ to \$L(theta mid x)\$ i.e. \$L:Thetatimes{cal X} rightarrow mathbb{R} \$. the random function \$L(cdot mid X)\$ we could also consider that the likelihood is only the "observed" likelihood \$L(cdot mid x^{text{obs}})\$ in practice the likelihood … Read more

## Solved – Likelihood ratio test for comparing two exponential distributions

I am trying to use a likelihood ratio test to compare the parameters of two exponential distributions. by this thread Likelihood Ratio for two-sample Exponential distribution I found that I can use \$\$ frac {L_{H_1}(hat theta_1,,hat theta_2)}{L_{H_0}(hat theta_0)} = frac {(hat theta_0)^{n_1+n_2}}{(hat theta_1)^{n_1}(hat theta_2)^{n_2}}=left(frac {hat theta_0}{hat theta_1}right)^{n_1} cdot left(frac {hat theta_0}{hat theta_2}right)^{n_2} \$\$ to calculate … Read more

## Solved – Likelihood ratio test for comparing two exponential distributions

I am trying to use a likelihood ratio test to compare the parameters of two exponential distributions. by this thread Likelihood Ratio for two-sample Exponential distribution I found that I can use \$\$ frac {L_{H_1}(hat theta_1,,hat theta_2)}{L_{H_0}(hat theta_0)} = frac {(hat theta_0)^{n_1+n_2}}{(hat theta_1)^{n_1}(hat theta_2)^{n_2}}=left(frac {hat theta_0}{hat theta_1}right)^{n_1} cdot left(frac {hat theta_0}{hat theta_2}right)^{n_2} \$\$ to calculate … Read more

## Solved – How to combine multiple prior components and a likelihood

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component 1". There are also two previous studies very similar to my own. One of them had an effect of 5 +/- 1 (prior component 2), … Read more

## Solved – How to combine multiple prior components and a likelihood

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component 1". There are also two previous studies very similar to my own. One of them had an effect of 5 +/- 1 (prior component 2), … Read more

## Solved – What’s the difference between likelihood and confidence in claim being true

I'm reading the IPCC report on climate change from 2007. In their uncertainity guide they make a distinction between likelihood and levels of confidence. What's the difference between the terms? http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-uncertaintyguidancenote.pdf Best Answer Likelihood, as M. Chernick wrote, has a very standard statistical meaning. "Levels of confidence" suggest a number of statistical concepts, but are … Read more