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There ain’t no such thing as a free lunch, at least according to the popular adage. Well, not anymore! Not when it comes to neural networks, that is to say. Read on to see how to improve your network’s performance with an incredibly simple yet clever trick called the Monte Carlo Dropout.

#bayesian-statistics #deep-learning #machine-learning #neural-networks

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onte Carlo Simulation is based on repeated random sampling. The underlying concept of Monte Carlo is to use **randomness **to solve problems that might be deterministic in principle. Monte Carlo simulation is one of the most popular techniques to draw inferences about a population without knowing the true underlying population distribution. This sampling technique becomes handy especially when one doesn’t have the luxury to repeatedly sample from the original population. Applications of Monte Carlo Simulation range from solving problems in theoretical physics to predicting trends in financial investments.

Monte Carlo has 3 main usages: estimate parameters or statistical measures, examine the properties of the estimates, approximate integrals

This article is about these 3 usages of the Monte Carlo procedures and about 3 Monte Carlo variants, statistical sampling techniques, which can be used to generate independent random samples. The article will cover the following topics:

```
- Introduction to Monte Carlo Simulation
- MC Parameter Estimation
- MC Examining the Estimate Properties
- MC Integrals Approximation
- Importance Sampling
- Rejection Sampling
- Inverse Transform Sampling
```

_This article is suited for readers who have prior Statistical knowledge since it will cover medium-level statistical concepts and examples. If you want to learn essential statistical concepts from scratch, you can check my previous article about _*Fundamentals Of Statistics here.*

#machine-learning #programming #monte-carlo #data-science #python #monte carlo simulation and variants with python

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We will provide examples of how you solve integrals numerically in Python. Let’s recall from statistics that the mean value can be calculated as.

**This implies that we can find an approximation of an interval by calculating the average value times the range that we integrate.**

Let’s say that we want to calculate the following integral where from WolframAlpha we get the solution:

#python #monte-carlo #numerical-methods #statistics

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Monte Carlo Simulations are an incredibly powerful tool in numerous contexts, including operations research, game theory, physics, business and finance, among others. It is a technique used to understand the impact of risk and uncertainty when making a decision. Simply put, a Monte Carlo simulation runs an enourmous amount of trials with different random numbers generated from an underlying distribution for the uncertain variables.

Here, we will dive into how to predict stock prices using a Monte Carlo simulation!

**What do we need to understand before we start?**

We want to predict the price of the stock today. We know the price of the stock yesterday. So, what’s missing? We obviously do not know the daily return that the stock is going to yield today. This is where Monte Carlo comes in! But first… how can we estimate the return?

**How do we predict the daily return of the stock? Brownian Motion.**

Brownian motion will be the main driver for estimating the return. It is a stochastic process used for modeling random behavior over time. Brownian motion has two main components:

- Drift — the direction that rates of returns have had in the past. That is, the expected return of the stock. You may ask yourself: why is the variance multiplied by 0.5? Becasue historical values are eroded in the future.
- Volatility — the historical volatility multiplied by a random, standard normal variable.

Together, these compute the brownian motion — ie the daily return of a stock!

This technique will be used for every day into the future you want to predict, and for however many trials the monte carlo simulation will run! Let’s get to coding!

#stock-market #forecasting #data-science #monte-carlo

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I became frustrated with the numbers not making sense on my Student Loan statements and speculative opinions about why you should not try to pay it off quicker if you are able to. I wanted to gauge my likelihood of paying off the full loan and practice my Data Science skills, so I simulated my earnings using Monte Carlo written in Python. Here I give a walk-through and evaluation of the findings.

Full code available on my GitHub.

Skip to **Method** and **Results** if you’re short of time!

In the UK, students can take out university loans for both tuition fees and living costs with Student Finance England, who provide loans underwritten by the Student Loans Company. Once you enter employment as a graduate, you then begin to pay off your loan on a monthly schedule as part of your tax deductions. Crucially, if you do not pay off your loan, plus interest, in 30 years, your loan is cancelled. They’re also substantial — approximately £42,000 one year after graduating in many cases.

Example of the statement summary graduates receive from the Student Loans Company after graduating. (Image by author).

- I found documentation surrounding repayment murky and dense. It’s confusing, and no-one online or in my circle could give me clear answers.
**Conflicting opinions**surrounding repayment— _“it comes out of my tax so I don’t even notice it” _and*“most students won’t pay it off and it will get cancelled, so paying extra upfront might be wasting your money”*— but this paints “most students” with a broad brush, which spans a huge variance in earnings (and therefore amortisation) between subjects, university attended and industry.- I wanted to cut through the noise and capture the dynamics between the accruing interest and monthly tax repayments to quantify my own probability of paying it off.

1. How likely am I to pay off my student loan inside of the 30-year payment window, given my probable salary progression?

2. If you manage to save money, or come across a sum of money (e.g. inheritance), should you pay off some or all of your loan in the form of voluntary payments?

- Students taking out loans since 2012 are on “Plan 2” loans, and they are split up into instalments which you receive throughout your degree.
- Graduates only start paying back towards their loan when they earn more than £2,214 a month or £26,568 a year (pre-tax) as of the date this article was published.
- Loan repayments are deducted from your salary when you are paid by your employer (in the UK, usually the last working day of the month) — these are due the April after graduating if you earn above the threshold.
- You begin accruing interest from the day of your first loan instalment (i.e., shortly after arriving at university).
- Interest accrues daily.
- Plan 2 loans are
**written off**30 years after the April you were first due to repay if you are still repaying.

- The amount you pay back on your Student Loan is calculated as a percentage of how much you earn above a threshold, so it behaves much more like a tax, rather than regular loan repayments.
- Interest rates are variable, not fixed — they are calculated as the RPI (Retail Price Index) measure of inflation, plus 3%, so rates have been as high as 6.6% in recent years.
- Interest rates change each academic year (September 1st), using the RPI inflation for the financial year ending in the previous March (i.e., March 2019 RPI is used for the rate set on September 1st 2020).

#data-science #programming #monte-carlo #student-loans #personal-finance #python

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Learn how to use Monte Carlo Simulation in Finance with Python to optimize the portfolio and visualize the result.

We start by exploring how Monte Carlo Simulation works with simple examples. This leads towards an understanding how useful it is to optimize your portfolio using the Sharpe Ratio.

#python #monte carlo