## Sunday, February 9, 2014

### The BayesFactor package

This blog is a companion to the BayesFactor package in R (website), which supports inference by Bayes factors in common research designs. Bayes factors have been proposed as more principled replacements for common classical statistical procedures such as $p$ values; this blog will offer tutorials in using the package for data analysis.
In this first post, I describe the general logic of Bayes factors using a very simple research example. In the coming posts, I will show how to do a more complete Bayesian data analysis using the R package.

### What is a Bayes factor?

Suppose that two researchers are interested in public opinion about public smoking bans. Paul believes that 70% of the public support such bans; Carole believes that the support is less, at 60%. Paul and Carole decide ask 100 randomly selected people whether they support public smoking bans.
Because Paul and Carole have very specific beliefs about the true support for the public smoking bans, they also have predictions about how the sample will turn out. Of course, Paul's best guess is that 70 out of 100 will support smoking bans, and likewise Carole's best guess is 60. However, Paul and Carole can be more specific because the Binomial distribution is appropriate for model the kind of random sample. The figure below shows the predictions of Carole (blue) and Paul (red).

Having made predictions, Paul and Carole collect their random sample. Of the 100 people in the sample, 62 are supportive. This seems to support Carole because the observation is closer to Carole's average prediction; however, Paul points out that he also predicted that 62 was possible, and so his hypothesis is not ruled out. It does seem obvious that the observation supports Carole, but by how much?
One way of answering this question is to ask about the relative weight of the evidence: that is, how convincing is the observation with respect to the two hypothesis in question? In order to answer this question, we need a way of understanding what it means for evidence to change what we believe. We start with the idea of prior odds, which describe the degree to which we favor one hypothesis over another before we see the data. These can be written as $\frac{P(\cal H_c)}{P(\cal H_p)}$ where $P$ here represents plausibility and $\cal H_c$ and $\cal H_p$ are hypotheses of Carole and Paul. We can also describe the posterior odds, which describe the degree to which we favor one hypothesis over another after observing the data. $\frac{P(\cal H_c\mid y)}{P(\cal H_p\mid y)}$ where $y$ is what we observed. The question now is how we move from the prior odds to the posterior odds in the right way. Obviously, we cannot simply do this in any way we like. There must be some ways that are better than others. Bayes' rule gives a way to answer this question. Bayes' rule says that the relative plausibility of two hypotheses must be changed by the data in a particular way. $\frac{P(\cal H_c\mid y)}{P(\cal H_p\mid y)} = \frac{P(y\mid \cal H_c)}{P(y\mid\cal H_p)}\times\frac{P(\cal H_c)}{P(\cal H_p)}$ The term in the middle is a factor that we multiply the prior odds by to become the posterior odds. It represents how our relative beliefs should change in light of the data, and is called the Bayes factor. Luckily, the logic is quite simple:

The Bayes factor is the relative evidence in the data. The evidence in the data favors one hypothesis, relative to another, exactly to the degree that the hypothesis predicts the observed data better than the other.
We can now easily compute the evidence in the data favoring Carole over Paul. Carole specified that the observation $y=62$ had a probability of 0.0754. Paul specified that the observation $y=62$ had a probability of 0.0191. Under Carole's hypothesis, the observed data is much more likely than under Paul's, as shown in the figure below:

The Bayes factor — the evidence in the data — is precisely factor by which Carole's line is taller than Paul's, which is:
dbinom(y, N, carole)/dbinom(y, N, paul)

## [1] 3.953

or about 4. The observed data favors Carole by a factor of 4.
The figure below shows how much the data favors Carole for many possible observations. The data favor Carole when the Bayes factor is greater than 1, which happens for all observations 65 or less; the data favor Paul when the Bayes factor is less than 1, which happens when the observation greater than 65. For very large or small observations, the evidence is strong; for observations around 65, the evidence is slight, making it difficult to distinguish between the two hypotheses.

### Bayes factors when the parameter is uncertain

The above example was adequate for a simple demonstration, but lacks an important feature common to research: uncertainty about the true parameter value. Carole and Paul both had very specific hypotheses. Typically, this is not the case. Unless a hypothesis is very specific about a particular parameter (for instance, that an experimental manipulation has no effect), hypotheses are more diffuse.
Carole and Paul might have hypotheses that look something like the figure below:

Note that on the $x$-axis is the true proportion of the population who support the smoking bans. Suppose we call the true proportion of supporters $\theta$. Instead of believing that the true proportion is exactly 60%, Carole now believes that it is “around” 60%. The blue distribution shows how her belief drops off as the proportion is further away from 0.6. Likewise, Paul has adopted a distribution around 0.7. We refer to these curves as $p_c(\theta)$ for Carole and $p_p(\theta)$ for Paul.
Since the Bayes factor is determined by the probability that each hypothesis assigns to the observed data, we must determine what Paul and Carole would now predict. Carole's hypothesis $\cal H_c$ is a distribution of values of $\theta$, each weighted differently. Some values, like 0.6, are quite plausible; other values, like 0.7, are implausible. In order to determine the plausibility of a particular observation, we can determine the probability that it would occur given a particular value for $\theta$ (that is, $P(y\mid\theta)$). We then weight this probability by the plausibility given to that particular $\theta$ value, $p(\theta)$. For each hypothesis, this can be represented by an integral: $P(y\mid{\cal H}_c) = \int P(y\mid \theta)p_c(\theta)\,d\theta$ where $p_c(\theta)$ is the function representing Carole's hypothesis. The probability of an observation is thus a weighted average over all the different possible values for $\theta$. The figure below shows how the more diffuse hypothesis affects the predictions for data. In gray are the predictions from the old, single point hypothesis. In color are the new predictions from the diffuse hypothesis. The diffuse hypotheses have made the predictions for the data more diffuse. The bottom row shows the new predictions made by Carole and Paul side-by-side.

We can now compare the two hypotheses in light of the observation of 62 supporters out of 100, as the figure below shows. The Bayes factor has been reduced to 2.8267 in favor of Carole. This makes sense; both Carole and Paul have hedged their predictions for the data, making them more spread out and in the process less distinct. The reduction in the Bayes factor reflects this by becoming attenuated.

Uncertain hypotheses are not a problem for the Bayes factor to handle, as long as they can be specified as probability distributions, but hedging one's bets with more uncertain predictions will limit reduce the amount of evidence that can be accumulated in favor of one's hypothesis. This means Bayes factors penalizes flexible hypotheses, yielding a natural Occam's razor.

### Strength of the evidence

One might ask how much evidence is provided by the Bayes factors of 3.9534 and 2.8267 found above. The simple answer is that the number itself is directly interpretable, since it arises as the shift of relative odds. In this, the Bayes factor is different from other proposed (but ultimately flawed) evidential statistics, such as the $p$ value, because such statistics have no direct evidential interpretation.
Nevertheless, various labels for the “strength” of a Bayes factor have been proposed. Kass and Raftery (1995), for instance, propose that Bayes factors between 1/3.2 and 3.2 are “not worth more than a bare mention”. See the Wikipedia entry for Bayes factors for more details.

### Summary

• Bayes factors represent the weight of evidence in the data for competing hypotheses. Bayes factors are the degree to which the data shift the relative odds between two hypotheses. There are principled reasons why we should interpret the Bayes factor as a measure of the strength of the relative evidence.
• The Bayes factor is intimately linked to the predictions of a hypothesis. Because the evidence for a hypothesis given by the degree to which it has predicted the observed data, hypotheses that do not have predictions for data cannot accumulate evidence. Hypotheses with no predictions, such as $\mu\neq0$ are not allowed.
• The Bayes factor can be directly interpreted, without recourse to labels. The strength of the Bayes factor is reflected by the fact that it is a multiplicative change in odds. However, some authors provide labels to help interpret evidence.
In the next post, we will discuss Bayes factors for one-sample designs with the BayesFactor package.

1. Thanks so much for these great posts.

Is there an example of calculating a Bayes Factor for the difference of proportions as described in the example above? Can this be done in the BayesFactor package?

Thanks!

1. Hi Max, I don't have a blog post about that yet, but Bayes factors of the sort you want can be calculated using the contingencyTestBF() function. See the manual here for an example: http://bayesfactorpcl.r-forge.r-project.org/#ctables

2. Seu artigo reflete o dom maravilhoso que você possui em mostrar com uma clareza cristalina conceitos que em outros textos jamais encontramos. Continue escrevendo artigos pois desse modo vc desmistifica conceitos que parecem complexos em conceitos tão inteligíveis quanto grande é a sua inteligência didática.

2. Hi Richard, this post is genial. The Bayes factor is common tool in phylogenetic research to decide between two evolutionary hypotheses (models).
This is calculated as BF= 2(harmonic mean 1(complex model) - harmonic mean2 (simple model).
Do you know this procedure?
Best regards

1. Oops, missed the reply. See response below.

3. Hi Violeta, yes, I am aware of the harmonic mean estimator. BayesFactor does not use that estimator; instead, it uses importance sampling, simple Monte Carlo sampling, Gaussian quadrature, and a Laplace approximation (depending on the situation). They all estimate the same quantity, but the harmonic mean estimate is notoriously unstable (see Radford Neal's comment here).

4. Hi,
Thank you for this post. Do you have the R code for this example? I downloaded the package and I didn't see an example like this one.

5. Thank you so much for posting this article. It helped me a lot.

In the last paragraph, you mentioned that "The Bayes factor is intimately linked to the predictions of a hypothesis."

Then how can I get the predictive density of x given a specific hypothesis?
I guess p(x|H) is not a predictive probability density, but just a probability that gives us information about the plausibility of observed data x under H...

6. Thank-you for your nice post Richard!

Could you give further detail about how you obtained the 2.8267 result for the more diffuse hypotheses? In particular, what type of distributions are those shown in the first figure of the section "Bayes factors when the parameter is uncertain"? If I am not mistaken those are the priors, right?

Thanks

7. So far the best explanation (to me) that I came across, thanks very much Richard!

8. This comment has been removed by the author.

9. Thank you so much for posting this article. It helped me a lot.

10. Great article Glad to find your blog. Thanks for sharing.
ibm-message-broker training in chennai

11. • Very good effort in collecting information.........
websphere training in chennai

12. • great article!!!!!This is very importent information for us.I like all content and information.I have read it.You know more about this please visit again.

oracle training in chennai

13. Hello Mr Richard,
Thank you for your posts. I don't really understand much since i'm not good at maths.
Can you help me simplify the calculation of bayes factor from the following problem?

I run Beast using HIV dataset testing two clock model Relax Clock and Strict Clock. So from result of log file from the Beast which can views in tracer, how can i calculate bayes factor to compare these two model?
Thank you in advance for your help.
Sophearot (from Cambodia)

14. Also there should be high amount of transparency as to where the software stands in terms of quality, field service software

15. Testing makes your customer to get full satisfcation on your service since it found out all the bugs and errors
and rectify it. Selenium is the best tool ever to test web based applications. Thank you for your information.

Regards:

Selenium Training in chennai |
Selenium Training

16. Can I just say what a relief to discover somebody that really knows what they're discussing online. You definitely understand how to bring a problem to light and make it important. You can review my latest post about ms office More people should check this out and understand this side of your story. I can't believe you are not more popular given that you definitely possess the gift.

18. Get intopc for best pc software you can get here

19. Gta 5 Mobile Dwgamez for best pc software you can get here

20. This comment has been removed by the author.

21. This comment has been removed by the author.

22. Hey, Thanks for such a nice share. You can download windows 7 lite

23. Thanks for sharing this details.

Online ERP Software

24. The very next time I read a blog, Hopefully it does not fail me just as much as this one. I mean, Yes, it was my choice to read, nonetheless I really believed you would probably have something helpful to talk about. All I hear is a bunch of moaning about something you could possibly fix if you weren't too busy seeking attention.
Get the Windows 10 Gamer Edition

25. What are the features of the kindle? This article talks about top six features of the wireless reading device, Kindle Fire. Kindle adbfire download enables user to surf the internet, listen to music, watch movie, read books and etc. Here are the top six features of the Kindle Fire that I am going to share with you.

26. I found this informative and interesting blog so i think so its very useful and knowledge able.I would like to thank you for the efforts you have made in writing this article.
angularjs Training in marathahalli

angularjs interview questions and answers

angularjs Training in bangalore

angularjs Training in bangalore

angularjs Training in chennai

automation anywhere online Training

27. Good Post! Thank you so much for sharing this pretty post, it was so good to read and useful to improve my knowledge as updated one, keep blogging.
python course in pune
python course in chennai
python course in Bangalore

28. I am sure this post has helped me save many hours of browsing other related posts just to find what I was looking for. Many thanks!

Core Java interview questions and answers

Java training in Chennai | Java training in Tambaram

Java training in Chennai | Java training in Velachery

Java training in Chennai | Java training in Omr

29. Inspiring writings and I greatly admired what you have to say , I hope you continue to provide new ideas for us all and greetings success always for you..Keep update more information.

rpa training in chennai
rpa training in bangalore
rpa course in bangalore
best rpa training in bangalore
rpa online training

30. Nice tutorial. Thanks for sharing the valuable information. it’s really helpful. Who want to learn this blog most helpful. Keep sharing on updated tutorials…
python training institute in marathahalli
python training institute in btm
Python training course in Chennai

31. It's wonderful commands like mind relaxes with understand words of knowledge by information's. check out https://www.ayevainstitute.com/

32. I am really enjoying reading your well written articles.
It looks like you spend a lot of effort and time on your blog.Keep Doing.
Digital Marketing Training in Bangalore
German Language Training in Bangalore
Best German Classes in Bangalore
German Coaching Classes in Bangalore

33. Superior post, keep up with this exceptional work. It's nice to know that this topic is being also covered on this web site so cheers for taking the time to discuss this! Thanks again and again! Adobe Flash CS3 Professional Free Download

34. Download your favorite Sinhala songs with Sindu. So, you can get Lyrics, Chords and Sinhala sindu, Sindu Potha, Sindu Sinhala MP3 for free, hundreds of new songs published everyday!

35. Nice and interesting information and informative too.
Can you please let me know the good attraction places we can visit: Bangalore to Chennai

36. Good Post! Thank you so much for sharing this pretty post, it was so good to read and useful to improve my knowledge as updated one, keep blogging
samsung mobile service center in chennai
samsung mobile service center
samsung mobile service chennai
samsung mobile repair
samsung mobile service center near me
samsung service centres in chennai

38. if you want to download artcut 2009 from rahim-soft

39. if you want to get information please go to high-program

40. who need CorelDRAW 2019 portable please go to Rahim-portable

42. if you want to download filemora 9.0 from getmaczone

43. Attend The Python training in bangalore From ExcelR. Practical Python training in bangalore Sessions With Assured Placement Support From Experienced Faculty. ExcelR Offers The Python training in bangalore.
python training in bangalore

44. This comment has been removed by the author.

45. Getting a business cash advance is simple and easy for most small businesses, and even those who have poor credit scores. While this does not apply to bank loans, these are the requirements of private lenders, and private lenders are amongst the leading funders at this time. Most business owners who https://adobe.cheapsoftwaredownload.net/ looking for funding and are unaware of the current requirements and developments of the financial sector, visit their local bank

46. I learned World's Trending Technology from certified experts for free of cost. I Got a job in decent Top MNC Company with handsome 14 LPA salary, I have learned the World's Trending Technology from hadoop training in btm experts who know advanced concepts which can help to solve any type of Real-time issues in the field of Hadoop. Really worth trying

47. It’s great to come across a blog every once in a while that isn’t the same out of date rehashed material. Fantastic read.
Data science Course Training in Chennai |Best Data Science Training Institute in Chennai
RPA Course Training in Chennai |Best RPA Training Institute in Chennai

48. This comment has been removed by the author.

49. I feel really happy to have seen your webpage and look forward to so many more entertaining times reading here. Thanks once more for all the details.
Rpa training in chennai | RPA training class in chennai

50. Visit for Python training in Bangalore:- Python training in Bangalore

51. Pretty good post. I just stumbled upon your blog and wanted to say that I have really enjoyed reading your blog posts. Any way I’ll be subscribing to your feed and I hope you post again soon.
360digitmg data science training

52. I am looking for and I love to post a comment that "The content of your post is awesome" Great work!

data science training

53. Writing with style and getting good compliments on the article is quite hard, to be honest.But you've done it so calmly and with so cool feeling and you've nailed the job. This article is possessed with style and I am giving good compliment. Best!
360 digitmg course on big data analytics

54. Such a very useful Blog. Very interesting to read this article. I have learn some new information.thanks for sharing. Click here for data science course in pune with placements

55. Someone Sometimes with visits your blog regularly and recommended it in my experience to read as well.

360digitmg IOT Training Course

56. Awesome blog. I enjoyed reading your articles. This is truly a great read for me. I have bookmarked it and I am looking forward to reading new articles. Keep up the good work!
agile course

57. Awesome blog. I enjoyed reading your articles. This is truly a great read for me. I have bookmarked it and I am looking forward to reading new articles. Keep up the good work!
big data analytics course

58. I want to post a remark that "The substance of your post is amazing" Great work.

360digitmg Data Science Course in Malaysia

59. It is really explainable very well and i got more information from your site.Very much useful for me to understand many concepts and helped me a lot.ServiceNow training in bangalore

60. Very useful and information content has been shared out here, Thanks for sharing it.Mulesoft training in bangalore

61. This is a great article thanks for sharing this informative information. I will visit your blog regularly for some latest post. I will visit your blog regularly for Some latest post.big data analytics courses

62. A debt of gratitude is in order for ExcelR Data Analytics Course In Pune the blog entry amigo! Keep them coming...

63. A debt of gratitude is in order for sharing the information, keep doing awesome... I truly delighted in investigating your site. great asset...
Please check ExcelR Data Science Courses

64. Great post!I am actually getting ready to across this information,i am very happy to this commands.Also great blog here with all of the valuable information you have.Well done,its a great knowledge.dot net training in bangalore

65. Enjoyed reading the article above, really explains everything in detail, the article is very interesting and effective. Thank you and good luck…

Start your journey with Database Developer Training in Bangalore and get hands-on Experience with 100% Placement assistance from experts Trainers @Bangalore Training Academy Located in BTM Layout Bangalore.

66. That was really a great Article.Thanks for sharing information. Continue doing this.

Best SAP EWM Training in Bangalore - Learn from best Real Time Experts Institutes in Bangalore with certified experts & get 100% assistance.

67. Excellent Post. Learn here how Home elevators technology is Evolved, Vacuum elevator | Home lift

68. Great Post. Excellent and Very Good Informations. Home lift

69. Wonderful thanks for sharing an amazing idea. keep it...

Get Best SAP ABAP Training in Bangalore from Real Time Industry Experts with 100% Placement Assistance in MNC Companies. Book your Free Demo with Softgen Infotech.

70. I can’t imagine that’s a great post. Thanks for sharing.

Looking for SAP HANA ADMIN Training in Bangalore, learn from eTechno Soft Solutions SAP HANA ADMIN Training on online training and classroom training. Join today!

71. Very interesting blog and it is useful to all agile learners. Thanks for sharing the valuable info. To learn more about agile you can visit this blog.

72. Amazing knowledge and I like to share this kind of information with my friends and hope they like it they why I dorobotic process automation course

73. I wanted to thank you for this great read!! I definitely enjoying every little bit of it I have you bookmarked to check out new stuff you post. Project Management Software

74. We as a team of real-time industrial experience with a lot of knowledge in developing applications in python programming (7+ years) will ensure that we will deliver our best in python training in vijayawada. , and we believe that no one matches us in this context.

75. Shopclues winner list 2020 here came up with a list of offers where you can win special shopclues prize list 2020 by just playing a game & win prizes.
Shopclues prize list 2020
Shopclues lucky draw
Shopclues lucky draw 2020
Shopclues prize department
Shopclues prize department 2020

76. Snapdeal Winner List here came up with an Offer where you can win Snapdeal prize list 2020 by just playing a game & win prizes.
Snapdeal winner name2020 also check the Snapdeal lucky draw2020

77. this is good blog. thank you for the information
swalih.websume.in

78. Hello… DropshipDragon provides dropping for quality, affordable products direct from China to your customers. Perfect for eBay sellers and website owners alike!… 3d coat zbrush

79. We as a team of real-time industrial experience with a lot of knowledge in developing applications in python programming (7+ years) will ensure that we will deliver our best in python training in vijayawada. , and we believe that no one matches us in this context.

80. Your articles are inventive. I am looking forward to reading the plethora of articles that you have linked here. Thumbs up! accounting system Malaysia

You can also download more driver for your PC & Mac from below link:
https://softbuff.com/category/pc-driver/
https://softbuff.com/category/mac-driver/

This is the best YouTube channels

82. Pretty article! I found some useful information in your blog, it was awesome to read, thanks for sharing this great content to my vision, keep sharing.

aws course in bangalore
aws tutorial videos

83. Thank u so much
Thanks for sharing my web
http://alltopc.com

84. Thanks for the informative article About Angular Js. This is one of the best resources I have found in quite some time. Nicely written and great info. I really cannot thank you enough for sharing.
Java training in chennai | Java training in annanagar | Java training in omr | Java training in porur | Java training in tambaram | Java training in velachery

85. Good blog!!! It is more impressive... thanks for sharing with us... nice t see
'CCC Service'

86. thank you for sharing useful information
AWS Training in bangalore

87. Forex Signals, MT4 and MT5 Indicators, Strategies, Expert Advisors, Forex News, Technical Analysis and Trade Updates in the FOREX IN WORLD

Forex Signals Forex Strategies Forex Indicators Forex News Forex World

88. Really nice and interesting post. I was looking for this kind of information and enjoyed reading this one. Keep posting. Thanks for sharing.

data science using python and r programming vizag

89. Very nice blogs!!! i have to learning for lot of information for this sites...Sharing for wonderful information.Thanks for sharing this valuable information to our vision. You have posted a trust worthy blog keep sharing, data science course in Hyderabad

90. I am reading your post from the beginning, it was so interesting to read & I feel thanks to you for posting such a good blog, keep updates regularly.i want to share about websphere online training .

91. Very good points you wrote here..Great stuff...I think you've made some truly interesting points.Keep up the good work.
data science using python and r programming Guwahati

92. This is a great inspiring article.I am pretty much pleased with your good work.You put really very helpful information. Keep it up. Keep blogging. Looking to reading your next post. best email extractor

93. This is a great inspiring article.I am pretty much pleased with your good work.You put really very helpful information. Keep it up. Keep blogging. Looking to reading your next post. best email extractor

94. This is my first time i visit here and I found so many interesting stuff in your blog especially it's discussion, thank you. Yahoo Scraper

95. This comment has been removed by the author.

96. A Chatbot Development is a product program for reproducing wise discussions with human utilizing rules or man-made brainpower. Clients connect with the Chatbot development service by means of conversational interface through composed or spoken content. Chatbots can live in informing stages like Slack, Facebook Messenger bot developer and Telegram and fill some needs – requesting items, thinking about climate and dealing with your fund in addition to other things. As a Chatbot development company advancement organization our competency let you find happiness in the hereafter by taking care of clients all the more intelligently to accomplish wanted outcome. As a Chatbot companies we can streamline a large portion of your dreary undertakings, for example, voice bot advancement and client service, online business advices and so on.

97. That is really nice to hear. thank you for the update and good luck. B2B Lead Generation - Find new Sales Leads

98. Thanks for sharing informative post. Keep on sharing such useful information. Limo Hire Melbourne | Wedding Car Hire Melbourne

99. We have seen many blog but this the great one, Thanks for provide great informatic and looking beautiful blog, really nice required information & the things i never imagined and i would like to request, wright more blog and blog post like that for us. Thanks you once agian

birth certificate in delhi
birth certificate in noida
birth certificate in ghaziabad
birth certificate in gurgaon
birth certificate agent in delhi
marriage registration in delhi
marriage certificate delhi
how to change name in 10th marksheet
marriage registration in ghaziabad
marriage registration in gurgaon

100. Nice article.This post is very informative.Check this best python training in bangalore with placement

101. Good Post! Thank you so much for sharing this pretty post, it was so good to read and useful to improve my knowledge as updated one, keep blogging
DevOps Training in Chennai

DevOps Course in Chennai

102. Your blog has chock-a-block of useful information. I liked your blog's content as well as its look. In my opinion, this is a perfect blog in all aspects. Salesflow alternative

103. Make Our Moments present the best places in Kerala to visit. God’s own country has several treasures hidden in the different locations. It may be a hill station or a forest or in a waterfall. This rich biodiversity makes Kerala the No. 1 tourist destination in India. We will help you to find the best Kerala tour packages for you and your loved ones.Explore the strange appeal of Kerala’s hill stations and wildlife. With a four day tour in Munnar, houseboat, and Cochin you can enjoy the attractions in these places and have fun with all the sport and fun activities your destination provide. Munnar shows you the beauty of hills and hill town in Kerala with broad never-ending tea and spice plantations, view from the mountains, and waterfalls. Your journey in the houseboat takes you by the charm of the backwaters and forests while Kochi takes you through the culture, arts, and traditions.

104. Nice & Informative Blog !
If you are looking for the best accounting software that can help you manage your business operations. call us at QuickBooks Customer Support Number 1-855-974-6537.

105. Hey!! Great work. You have a very informative blog .You are doing well. Keep it up. We will also provide Quickbooks Subscription Error to alter Quickbook’s issues. If you have any issues regarding Quickbooks dial +1-8555-756-1077 for getting instant help.

106. Superb Information, I really appreciated with it, This is fine to read and valuable pro potential, I really bookmark it, pro broaden read. Appreciation pro sharing. I like it. ExcelR Data Analytics Courses

107. This was incredibly an exquisite implementation of your ideas ExcelR Data Analytics Courses

108. I was finding about this information. Finally saw your blog. Nice and thansk rewasd crack

109. I really appreciate your efforts for this Article and wonderful Knowledge. I want to share with you all a useful source Best Things to do in Havelock Island

110. Thank you so much for posting this article. It helped me a lot.
To apply for a visa, you can check countries online Pakistan tourist visa for the process can be done online using Pakistan's online visa system. Pakistan offers two types of tourist visas, namely tourist visa and tourist visa on arrival.
With this new system, processing of a e visa Turkey for tourist and business travel to Turkey has become simple and inexpensive as long as your application is completed.

111. ExcelR provides Business Analytics Course. It is a great platform for those who want to learn and become a Business Analytics Courses. Students are tutored by professionals who have a degree in a particular topic. It is a great opportunity to learn and grow.

112. wow, great, I was wondering how to cure acne naturally. and found your site by google, learned a lot, now i’m a bit clear. I’ve bookmark your site and also add rss. keep us updated.

Data Science Course in Mysore

113. Hey! Excellent work. Being a QuickBooks user, if you are struggling with any issue, then dial QuickBooks Phone Number (877)948-5867. Our team at QuickBooks will provide you with the best technical solutions for QuickBooks problems.

114. ExcelR provides data analytics courses. It is a great platform for those who want to learn and become a data analytics Course. Students are tutored by professionals who have a degree in a particular topic. It is a great opportunity to learn and grow.

data analytics courses
data analytics course

115. ExcelR provides data analytics courses. It is a great platform for those who want to learn and become a data analytics Course. Students are tutored by professionals who have a degree in a particular topic. It is a great opportunity to learn and grow.

data analytics courses
data analytics course

116. Nice Blog
QuickBooks is one of the best accounting software that is developed to manage all the accounting tasks of a business. In case you want quick solutions for QuickBooks problems, call us on QuickBooks Customer Service Number and get the most feasible solutions for QuickBooks errors.

117. Great writeup! Thanks so much for putting in the work and sharing with everyone.In case if you face any techincal issue in QuickBooks, you can contact Us:

QuickBooks Customer Service

118. Thanks for this :) Online Turkish visa for the application of citizens of Armenia issued by the Turkish government to enter and travel in Turkey. Apply for your Turkey visa for Armenia citizens online as it is a much more convenient way to obtain a visa.

119. I think it could be more general if you get a football sports activity. ExcelR Data Scientist Course In Pune

If you face any Error in QuickBooks then immediately contactQuickbooks Error Support

121. Amazing information. I was looking for this one. Thanks a lot. I have a suggestion for the Best Data Science Course in Gurgaon. If you want to enroll in Data Science Course, Join 99 Digital Academy as it offers an affordable Data Science Course. Click to Enroll Today.
Best Data Science Course in Gurgaon.

122. Thanks for sharing the informative article you can visit us at Python Courses Fees ,we are from educational backgroud

123. Hey, I like this and its helpful for me and i appreciate your work.
Buy Instagram Followers India

124. Wow, Great information and this is very useful for us. Golden Triangle Tour Package India

125. thanks for sharing the details

126. We at YES School Of Language Provide German Classes in chennai

127. Very interesting. Thanks for sharing

Cloud managed services are really important for a business. These service includes migration, maintenance, optimization and provide public, private, and hybrid cloud environment. We provide the best service for it.
Azure cloud management services!

128. Hey! What a wonderful blog. I loved your blog. QuickBooks is the best accounting software, however, it has lots of bugs like QuickBooks Error. To fix such issues, you can contact experts via QuickBooks Support Phone Number

129. Hey! Mind-blowing blog. Keep writing such beautiful blogs. In case you are struggling with issues on QuickBooks software, dial QuickBooks Customer Service Phone Number . The team, on the other end, will assist you with the best technical services.

130. Thankful to you for sharing this.
tatkal passport

131. Thanks For Sharing Such Content.
renew passport

132. Nice post thanks for sharing.

Tech Profuse offers complete Tableau integration solutions to help you quickly analyze, visualize and share information with accurate and engaging dashboards to drive better business insights.
Tableau Consulting Services

133. Nice Post and information.

Kerala, Gods own country is a small beautiful land covered with the greenery of nature. This amazingly beautiful state in India has always been a wonder for all. The enchanting beauty of exotic hill stations, sun kissed beaches, serene backwaters, dense green vegetation, huge variety of wild life, lagoons, canals, rivers and lakes along with pilgrim centres and several historical monuments increases the number of visitors coming for tour to Kerala.
Tour to Kerala

134. Hey there
Good work, Keep on posting
Visit - Elewayte - Best Online Certification courses to get trained in one of the best online training institutions with internships and placement assistance

135. We want to make every vacation and every moment a memorable experience! Come, discover a new way of travel - MaldivesPlanet
https://maldivesplanet.com/

136. Thanks for sharing post and Useful Information.
​Non ECR Passport
Passport fees In India

137. Azure Cloud Consulting Services : Techprofuse is one of the most trusted cloud managed services provider and end-to-end Cloud Consulting Services company in India.
Cloud Computing Services

138. Field Force Tracker provides lot of features such as work order management, parts and asset management, warranty and contract management, field service staff management and more.
Our Field Service software is used for HVAC, Electrical, Plumbing, Photocopiers, IT & Office Automation, Industrial Goods, Home Appliances, Renovations, and Generators & More.

139. Tata Sky, The leader in DTH with unmatched service offers your choice of channels and Packages at best offers. Get Best Deals on DTH at DTH Hyderabad.
Tata Sky Packages

140. what a informative and knowlegeable websites.Jihosoft File Recovery Activation Code

141. This impressed me so much amazing. Keep working and providing information
activator

142. Excellent article. Keep writing such kind of info on your site.
I am really impressed by it .You’ve done an incredible job. Keep it up if you want to progress in the future.
Full Version

143. Leawo Prof. Media is an amazing Post with good content.FIND CRACK is the best crack software site for all Mac and Windows users all over the world.

144. I’m glad that you just shared this helpful info with us. Please keep us informed like this
FL Studio Crack

145. Click this link